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Let $k$ be $\mathbb{C}$ (because that is fewer keystrokes in LaTeX). Let $\mathbb{P}^1_k$ be $\text{Proj}\ k[R,S]$. Let $\mathbb{P}^3_k$ be $\text{Proj}\ k[T_0,T_1,T_2,T_3]$. Let $e$ be a nonnegative integer. Let $b_0,b_1,b_2,b_3 \in k[R,S]_e$ be a $4$-tuple of homogeneous polynomials of degree $e$ on $\mathbb{P}^1_k$.

Let $\mu_5$ denote the group of $5^{\text{th}}$ roots of unity in $\mathbb{G}_{m,k}$. Let $\mu_5$ act on $\mathbb{P}^1_k\times_{\text{Spec}\ k}\mathbb{P}^3_k$ as follows, $$\zeta \bullet([S_0,S_1],[T_0,T_1,T_2,T_3]) = ([S_0,S_1],[T_0,\zeta T_1, \zeta^2 T_2, \zeta^3 T_3]).$$ The polynomial $f$ is invariant for this action. Thus, there is an induced action on $Y$.

Let $k$ be a field of characteristic different from $5$. Let $\mathbb{P}^1_k$ be $\text{Proj}\ k[R,S]$. Let $\mathbb{P}^3_k$ be $\text{Proj}\ k[T_0,T_1,T_2,T_3]$. Let $e$ be a nonnegative integer. Let $b_0,b_1,b_2,b_3 \in k[R,S]_e$ be a $4$-tuple of homogeneous polynomials of degree $e$ on $\mathbb{P}^1_k$.

Let $\mu_5$ denote the group $k$-scheme of $5^{\text{th}}$ roots of unity in $\mathbb{G}_{m,k}$. Since $\text{char}(k)$ is not $5$, this is $k$-smooth. Let $\mu_5$ act on $\mathbb{P}^1_k\times_{\text{Spec}\ k}\mathbb{P}^3_k$ as follows, $$\zeta \bullet([S_0,S_1],[T_0,T_1,T_2,T_3]) = ([S_0,S_1],[T_0,\zeta T_1, \zeta^2 T_2, \zeta^3 T_3]).$$ The polynomial $f$ is invariant for this action. Thus, there is an induced action on $Y$.

Since the morphism $\pi$ is $\mu_5$-invariant, there is a unique $k$-morphism, $$\pi':X'\to \mathbb{P}^1_k,$$ such that $\pi'\circ \nu$ equals $\pi$. By Lemma 3, the singular locus of $\pi'$ equals $X'_0\cup X'_1\cup X'_2 \cup X'_3$, for $X'_i:= \nu(Y_i)$. Thus, the smooth locus of $\pi$ equals the open complement $U$ of $\cup_i X'_i$ in $X'$. On $U$, the transitivity exact sequence of $\pi$ gives a short exact sequence, $$0 \to \pi^*\Omega_{\mathbb{P}^1/k}^1 \to \Omega_{U/k} \to \Omega_{\pi} \to 0.$$ This induces a short exact sequence, $$0 \to \pi^*\Omega_{\mathbb{P}^1/k}\otimes\Omega_{\pi} \to \Omega^2_{U/k} \to \Omega^2_{\pi} \to 0.$$ This also induces an isomorphism, $$\Omega^3_{U/k} \cong \pi^*\Omega_{\mathbb{P}^1/k}\otimes \Omega^2_{\pi}.$$

Since the global sections of $\Omega^r_\pi$ are zero for $r>0$, the only possibility for a nonzero global section of $\Omega^r_{U/k}$ is a global section of $\pi^*\Omega_{\mathbb{P}^1_k/k}$ when $r=1$. Of course this equals the restriction to $U$ of $\pi^*\mathcal{O}(-2)$. Since $X'$ is normal and since $\cup_i X'_i$ consists of finitely many points, every global section of $\pi^*\mathcal{O}(-2)$ on $U$ is the restriction of a unique global section on all of $X'$ of $\pi^*\mathcal{O}(-2)$. By the projection formula, every such global section is the zero global section. QED

Since the morphism $\pi$ is $\mu_5$-invariant, there is a unique $k$-morphism, $$\pi':X'\to \mathbb{P}^1_k,$$ such that $\pi'\circ \nu$ equals $\pi$. By Lemma 3, the singular locus of $\pi'$ equals $X'_0\cup X'_1\cup X'_2 \cup X'_3$, for $X'_i:= \nu(Y_i)$. Thus, the smooth locus of $\pi$ equals the open complement $U$ of $\cup_i X'_i$ in $X'$. On $U$, the transitivity exact sequence of $\pi$ gives a short exact sequence, $$0 \to \pi^*\Omega_{\mathbb{P}^1/k}|_U \to \Omega_{U/k} \to \Omega_{\pi}|_U \to 0.$$ This induces a short exact sequence, $$0 \to \pi^*\Omega_{\mathbb{P}^1/k}\otimes\Omega_{\pi}|_U \to \Omega^2_{U/k} \to \Omega^2_{\pi}|_U \to 0.$$ This also induces an isomorphism, $$\Omega^3_{U/k} \cong \pi^*\Omega_{\mathbb{P}^1/k}\otimes \Omega^2_{\pi}|_U.$$

Since the global sections of $\Omega^r_\pi$ on $U$ are zero for $r>0$, the only possibility for a nonzero global section of $\Omega^r_{U/k}$ is a global section of $\pi^*\Omega_{\mathbb{P}^1_k/k}$ with $r=1$. Of course $\pi^*\Omega_{\mathbb{P}^1_k/k}$ equals $\pi^*\mathcal{O}(-2)$. Since $X'$ is normal and since $\cup_i X'_i$ consists of finitely many points, every global section of $\pi^*\mathcal{O}(-2)$ on $U$ is the restriction of a unique global section on all of $X'$ of $\pi^*\mathcal{O}(-2)$. By the projection formula, every such global section is the zero global section. QED

One example arises as the total space of a family of Godeaux surfaces over $\mathbb{P}^1_k$ with sufficiently positive discriminant.

Notation 2. Denote by $f$ be the section of $\text{pr}_{\mathbb{P}^1}^*\mathcal{O}(e)\otimes \text{pr}_{\mathbb{P}^3}^*\mathcal{O}(5)$, $$f = b_0 T_0^5 + b_1 T_1^5 + b_2 T_2^5 + b_3 T_3^5.$$ Denote by $Y$ be the zero scheme of $f$ as a hypersurface in $\mathbb{P}^1_k\times_{\text{Spec}\ k} \mathbb{P}^3_k$. For each $i=0,1,2,3$, denote by $p_i$ the $k$-point of $\mathbb{P}^3_k$ where $T_j$ vanishes for every $j\neq i$. Also denote by $Y_i\subset Y$ the product $\text{Zero}(b_i)\times\{p_i\}$. Finally, denote the restrictions of $\text{pr}_{\mathbb{P}^1}$ and $\text{pr}_{\mathbb{P}^3}$ to $Y$ as follows, $$\pi:Y\to \mathbb{P}^1_k, \ \ \rho:Y\to \mathbb{P}^3_k.$$

Lemma 3. The hypersurface $Y$ is smooth. The singular locus of the projection $\pi$ equals $Y_0\cup Y_1\cup Y_2 \cup Y_3$. The projection $\rho$ is smooth over $p_i$ for every $i=0,1,2,3$. The dualizing sheaf of $Y$ equals the restriction of $\text{pr}_{\mathbb{P}^1}^*\mathcal{O}(e-2)\otimes \text{pr}_{\mathbb{P}^3}^*\mathcal{O}(1)$.

Proof. By the Jacobian criterion, the projection $\pi$ is smooth away from $\cup_i Y_i$. The fiber of $\rho$ over $p_i$ is the zero scheme of $b_i$, and this is reduced (hence smooth) by hypothesis. Thus, the projection morphism $\rho$ is smooth at every point of $\cup_i Y_i$. For every point of $Y$, either $\pi$ or $\rho$ is smooth, and hence $Y$ is everywhere smooth. The remaining computations are straightforward. QED

Let $\mu_5$ denote the group of $5^{\text{th}}$ roots of unity in $\mathbb{G}_{m,k}$. Let $\mu_5$ act on $\mathbb{P}^1_k\times_{\text{Spec}\ k}\mathbb{P}^3_k$ as follows, $$\zeta \bullet([S_0,S_1],[T_0,T_1,T_2,T_3]) = ([S_0,S_1],[T_0,\zeta T_1, \zeta^2 T_2, \zeta^3 T_3]).$$ The polynomial $f$ is invariant for this action. Thus, there is an induced action on $Y$.

Notation 4. Denote the quotient of this $\mu_5$-action on $Y$ by $\nu:Y\to X'$.

Lemma 5. The action of $\mu_5$ on $Y\setminus \cup_i Y_i$ is free. For every point of $\cup_i Y_i$, the age (in the sense of Reid -- Shepherd-Barron -- Tai) is $> 1$.

Proposition 6. The $k$-scheme $X'$ is projective, normal, $\mathbb{Q}$-Gorenstein, has ample $\mathbb{Q}$-canonical divisor class, and has only terminal finite quotient singularities. Finally, $X'$ is simply connected. Thus, every projective desingularization of $X'$ is a simply connected, projective $3$-fold of general type.

Proposition 7. For $r=1,2$, the only global section of $\Omega^r_\pi$ on $U$ is the zero section. For $r=1,2,3$, the only global section of $\Omega^r_{U/k}$ on $U$ is the zero section. Thus, for every projective desingularization $X$ of $X'$, the only global section of $\Omega^r_{X/k}$ on $X$ is the zero section.

Proof. These coherent sheaves are locally free sheaves on a smooth $k$-scheme. Every nonzero section on $U$ restricts to a nonzero section of the restriction on the generic fiber $F$ of $\pi$. The restriction of $\Omega^r_\pi$ to the generic fiber $F$ equals $\Omega^r_{F/k(\mathbb{P}^1)}$. The generic fiber $F$ of $\pi$ is a Godeaux surface over $k(\mathbb{P}^1)$. Thus, $\Omega^r_{F/k(\mathbb{P}^1)}$ has only the zero global section for $r=1$ and $r=2$.

One simply connected example arises as the total space of a family of Godeaux surfaces over $\mathbb{P}^1_k$ with sufficiently positive discriminant.

Notation 2. Denote by $f$ be the section of $\text{pr}_{\mathbb{P}^1}^*\mathcal{O}(e)\otimes \text{pr}_{\mathbb{P}^3}^*\mathcal{O}(5)$, $$f = b_0 T_0^5 + b_1 T_1^5 + b_2 T_2^5 + b_3 T_3^5.$$ Denote by $Y$ be the zero scheme of $f$ as a hypersurface in $\mathbb{P}^1_k\times_{\text{Spec}\ k} \mathbb{P}^3_k$. For each $i=0,1,2,3$, denote by $p_i$ the $k$-point of $\mathbb{P}^3_k$ where $T_j$ vanishes for every $j\neq i$. Also denote by $Y_i\subset Y$ the product $\text{Zero}(b_i)\times\{p_i\}$. Finally, denote the restrictions of $\text{pr}_{\mathbb{P}^1}$ and $\text{pr}_{\mathbb{P}^3}$ to $Y$ as follows, $$\pi:Y\to \mathbb{P}^1_k, \ \ \rho:Y\to \mathbb{P}^3_k.$$

Let $\mu_5$ denote the group of $5^{\text{th}}$ roots of unity in $\mathbb{G}_{m,k}$. Let $\mu_5$ act on $\mathbb{P}^1_k\times_{\text{Spec}\ k}\mathbb{P}^3_k$ as follows, $$\zeta \bullet([S_0,S_1],[T_0,T_1,T_2,T_3]) = ([S_0,S_1],[T_0,\zeta T_1, \zeta^2 T_2, \zeta^3 T_3]).$$ The polynomial $f$ is invariant for this action. Thus, there is an induced action on $Y$.

Notation 3. Denote the quotient of this $\mu_5$-action on $Y$ by $\nu:Y\to X'$. Denote by $\phi:X\to X'$ any strong desingularization of $X'$ that is projective.

Proposition 4. The projective $k$-scheme $X$ is simply connected and of general type. Moreover, for $r=1,2,3$, the only global section of $\Omega^r_{X/k}$ is the zero section.

This will be proved in stages.

Lemma 5. The hypersurface $Y$ is smooth. The singular locus of the projection $\pi$ equals $Y_0\cup Y_1\cup Y_2 \cup Y_3$. The projection $\rho$ is smooth over $p_i$ for every $i=0,1,2,3$. The dualizing sheaf of $Y$ equals the restriction of $\text{pr}_{\mathbb{P}^1}^*\mathcal{O}(e-2)\otimes \text{pr}_{\mathbb{P}^3}^*\mathcal{O}(1)$.

Proof. By the Jacobian criterion, the projection $\pi$ is smooth away from $\cup_i Y_i$. The fiber of $\rho$ over $p_i$ is the zero scheme of $b_i$, and this is reduced (hence smooth) by hypothesis. Thus, the projection morphism $\rho$ is smooth at every point of $\cup_i Y_i$. For every point of $Y$, either $\pi$ or $\rho$ is smooth, and hence $Y$ is everywhere smooth. The remaining computations are straightforward. QED

Lemma 6. The action of $\mu_5$ on $Y\setminus \cup_i Y_i$ is free. For every point of $\cup_i Y_i$, the age (in the sense of Reid -- Shepherd-Barron -- Tai) is $> 1$.

Lemma 7. The $k$-scheme $X'$ is projective, normal, $\mathbb{Q}$-Gorenstein, has ample $\mathbb{Q}$-canonical divisor class, and has only terminal finite quotient singularities. Finally, $X'$ is simply connected. Thus, every projective desingularization $X$ of $X'$ is a simply connected, projective $3$-fold of general type.

Lemma 8. For $r=1,2$, the only global section of $\Omega^r_\pi$ on $U$ is the zero section.

Proof. These coherent sheaves are locally free sheaves on the smooth $k$-scheme $U$. Every nonzero section on $U$ restricts to a nonzero section of the restriction on the generic fiber $F$ of $\pi$. The restriction of $\Omega^r_\pi$ to the generic fiber $F$ equals $\Omega^r_{F/k(\mathbb{P}^1)}$. The generic fiber $F$ of $\pi$ is a Godeaux surface over $k(\mathbb{P}^1)$. Thus, $\Omega^r_{F/k(\mathbb{P}^1)}$ has only the zero global section for $r=1$ and $r=2$. QED

Proof of Proposition 4. It only remains to prove that every global section of $\Omega^r_{X/k}$ is the zero section for $r=1,2,3$. Since $\phi$ is an isomorphism over $U$, it suffices to prove that every global section of $\Omega^r_{U/k}$ is the zero section.