As Akhil points out, you can have both types of behavior simultaneously. For instance, consider a curve $C$ that is a union of two irreducible components $C'$ and $C''$ that intersect transversally at a single point $p$ (smooth on each component). Let $C'$ be a smooth, genus $1$ curve. Let $C''$ be a nodal genus $1$ curve with geometric genus $0$. Let $\nu':C'\to B'$ be a double cover of a genus $0$ curve ramified at $p$. Let $\nu'':C''\to B''$ be a double cover of a genus $0$ curve ramified at $p$. Then there is a unique at-worst-nodal curve $B=B'\cup B''$ of genus $0$, glued together at $\nu'(p)=\nu''(p)$. Also there is a unique morphism $\nu:C\to B$ whose restriction to $C'$ is $\nu'$ and whose restriction to $C''$ is $\nu''$.
In some sense, this already shows how to produce explicit families: deform the pointed curve $(B,D)$, where $D$ is the union of the branch points of $\nu$ (not counting $p$). This can be made very explicit: first, identify $B$ with $(B'\times\{p\})\cup (\{p\}\times B'')$ inside $B'\times B''$. Now deform this curve in a general pencil of divisors $(\mathcal{B}_s)_{s\in S}$ in $B'\times B''$. The divisor $D$ is the intersection in $B'\times B''$ of $B$ and a divisor $E$ of type $(3,3)$. So also deform $E$ in a general pencil $(\mathcal{E}_t)_{t\in T}$ of divisors of type $(3,3)$ in $B'\times B''$. Now, over the base $S\times T$, consider the family of curves $\mathcal{B}_s$ with divisors $\mathcal{D}_{s,t} = \mathcal{B}_s \cap \mathcal{E}_t$.
The only issue is that, in order to explicitly form the double cover $\mathcal{C}_{s,t}\to \mathcal{B}_t$ branched over $\mathcal{D}_{s,t}$, we need an invertible sheaf on $\mathcal{B}_t$ and an explicit isomorphism of this invertible sheaf with $\mathcal{O}_{\mathcal{B}_t}(\mathcal{D}_{s,t})$. We can form this ideal sheaf on the generic fiber. However, filling in the branched cover almost certainly will require passing to a degree $2$ cover of $S\times T$ (at least, that is my recollection).