A $1$-dimensional connected algebraic subgroup of ${\mathbf G}_{\mathbf a}^n$ is isomorphic to ${\mathbf G}_{\mathbf a}^1$ if the ground field is perfect.
Are there any references concerning the structure of closed subgroups of ${\mathbf G}_{\mathbf a}^n$ ?
Does a closed subgroup of ${\mathbf G}_{\mathbf a}^n$ of dimension $k$ embedd in ${\mathbf G}_{\mathbf a}^{k}$ (for $K$ algebraically closed say) ?
If $K$ has characteristic $0$, then a closed subgroup of ${\mathbf G}_{\mathbf a}^n$ is a $K$-vector space, so the above obviously holds.
The answer to your question is no. Namely, take something like
$$\alpha_p=V(x^p-x)\subseteq\mathbf{G}_a$$
which is a zero-dimensional subgroup of $\mathbf{G}_a$ which does not embed in $\mathbf{G}_a^0$. A less stupid example can be obtained by considering something like $\mathbf{G}_a\times\alpha_p\subseteq\mathbf{G}_a^2$ which can't be embedded in $\mathbf{G}_a$.
As you pointed out, in characteristic $0$ every unipotent group is a vector group (i.e. isomorphic to a direct sum of $\mathbf{G}_a$). For all concerns related to unipotent groups you can see this book.