This may also be SGA3's example, since I am too lazy to go check, but my standard example of a reductive group degenerating to something solvable is to define a multiplication $\ast_t$ on $2 \times 2$ matrices by
$$\begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \ast_t \begin{pmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{pmatrix} = \begin{pmatrix} x_{11} y_{11} + t x_{12} y_{21} & x_{11} y_{12} + x_{12} y_{22} \\
x_{21} y_{11} + x_{22} y_{21} & t x_{21} y_{12} + x_{22} y_{22} \end{pmatrix}.$$
For $t \neq 0$, this is simply the ordinary $2 \times 2$ matrices under the change of coordinates $\left( \begin{smallmatrix} z_{11} & z_{12} \\ t z_{21} & z_{22} \end{smallmatrix} \right)$, so it is associative, and it gives a group when restricted to the open set $z_{11} z_{22} - t z_{21} z_{12} \neq 0$. I hope this shows you that there is nothing mysterious in the given formulas.
In the limit $t=0$, the above formulas still define an associative multiplication, which forms a group $G_0$ on the open set $z_{11} z_{22} \neq 0$. Explicitly,
$$\begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \ast_0 \begin{pmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{pmatrix} = \begin{pmatrix} x_{11} y_{11} & x_{11} y_{12} + x_{12} y_{22} \\
x_{21} y_{11} + x_{22} y_{21} & x_{22} y_{22} \end{pmatrix}.$$
Clearly, there is a map $G_0 \to \mathbb{G}_m^2$ projecting a matrix onto its diagonal entries. The kernel, matrices of the form $\left( \begin{smallmatrix} 1 & x_{12} \\ x_{21} & 1 \end{smallmatrix} \right)$, is clearly a copy of $\mathbb{G}_a^2$. So we have a short exact sequence $1 \to \mathbb{G}_a^2 \to G_0 \to \mathbb{G}_m^2 \to 1$ and $G_0$ is solvable.
As grghxy points out, every flat affine group scheme of finite type over a dvr embeds in some $GL_N$, so you are not adding anything with that condition.
This example can be modified to suit many tastes: Replace $z_{11} z_{22} - t z_{12} z_{21} \neq 0$ by $z_{11} z_{22} - t z_{12} z_{21} = 1$ if you like simple Lie groups better than reductive ones; quotient by $\pm \mathrm{Id}$ if you like your groups in adjoint form; work with $2 \times 2$ matrices and $\ast_t$ commutator if you like Lie algebras. Replace $t$ by $p$ in the above formulas to work over $\mathbb{Z}_p$.
