No. Recall that the Lusternik-Schnirelmann category of a space $X$, denoted $\operatorname{cat}(X)$, is the minumum $k$ such that $X$ may be covered by open sets $U_0,U_1,\ldots, U_k$ such that each inclusion is null-homotopic. The standard lower bound for LS-category is the cup-length of reduced cohomology: if $R$ is a commutative ring, and $x_1,\ldots, x_k\in \tilde{H}^*(X;R)$ are cohomology classes whose cup product $x_1\cdot \cdots \cdot x_k\in \tilde{H}^*(X;R)$ is non-zero, then $\operatorname{cat}(X)\leq k$. The proof of this is a nice exercise in the naturality of relative cup products.

Now suppose $N$ is a closed non-orientable $n$-manifold which is covered by two contractible charts $U_0$ and $U_1$. It follows that $\operatorname{cat}(N)=1$ (it can't be $0$, because closed manifolds are never contractible), and all cup products in $\tilde{H}^*(N;\mathbb{Z}/2)$ are trivial.

Since $N$ is non-orientable, its first Stiefel-Whitney class $w_1(N)\in H^1(N;\mathbb{Z}/2)$ is non-zero. Poincaré duality gives a non-singular pairing $H^1(N;\mathbb{Z}/2)\times H^{n-1}(N;\mathbb{Z}/2)\to H^n(N;\mathbb{Z}/2)$, in particular $w_1(N)\cdot y\neq 0$ for some $y\in H^{n-1}(N;\mathbb{Z}/2)$. Contradiction.

No. Recall that the Lusternik-Schnirelmann category of a space $X$, denoted $\operatorname{cat}(X)$, is the minumum $k$ such that $X$ may be covered by open sets $U_0,U_1,\ldots, U_k$ such that each inclusion is null-homotopic. The standard lower bound for LS-category is the cup-length of reduced cohomology: if $R$ is a commutative ring, and $x_1,\ldots, x_k\in \tilde{H}^*(X;R)$ are cohomology classes whose cup product $x_1\cdot \cdots \cdot x_k\in \tilde{H}^*(X;R)$ is non-zero, then $\operatorname{cat}(X)\geq k$. The proof of this is a nice exercise in the naturality of relative cup products.

Now suppose $N$ is a closed non-orientable $n$-manifold which is covered by two contractible charts $U_0$ and $U_1$. It follows that $\operatorname{cat}(N)=1$ (it can't be $0$, because closed manifolds are never contractible), and all cup products in $\tilde{H}^*(N;\mathbb{Z}/2)$ are trivial.

Since $N$ is non-orientable, its first Stiefel-Whitney class $w_1(N)\in H^1(N;\mathbb{Z}/2)$ is non-zero. Poincaré duality gives a non-singular pairing $H^1(N;\mathbb{Z}/2)\times H^{n-1}(N;\mathbb{Z}/2)\to H^n(N;\mathbb{Z}/2)$. In particular $w_1(N)\cdot y\neq 0$ for some $y\in H^{n-1}(N;\mathbb{Z}/2)$. Contradiction.