I bumped into the following question while working on a research problem on closed subschemes over a completed DVR. I hope someone could possibly answer it, or give a hint toward its resolution. To make the problem simple, let me state the relevant question in the following form.

Let $A$ be the local ring $\mathcal{O}_{\mathbb{A}^1_k, 0}= k[t]_{(t)}$ at the origin of the affine line over a field $k$ of characteristic $0$ and let $\widehat{A}=k[[t]]$ be its completion.

Let $I \subset \widehat{A} [y_1, \cdots, y_n]$ be a prime ideal of the ring of polynomial ring in $n$ variables with the coefficients in $\widehat{A}$. Let $Z$ be the closed subscheme given by this prime ideal $I$, which is in particular irreducible.

As in the title, I wonder if this "irreducibility" is a "continuous" property or a "limit" property, by which I mean, I wonder whether any of the following questions hold, in which case I would like to say "irreducibility is a continuous property in a sense".

(1) Under the above assumptions, is there a sufficiently large integer $N>0$ such that the image of the ideal, $\bar{I} \subset (\widehat{A}/(t^N))[y_1, \cdots, y_n]$ defines an irreducible scheme?

(2) Under the above assumptions, express $I= (f_1, \cdots, f_r)$ for some $f_i \in \widehat{A}[y_1, \cdots, y_n]$, which is possible because the ring is noetherian. For each integer $n>0$, let $f_{in}$ be the polynomial in $k[t][y_1, \cdots, y_n]$ obtained by ignoring the terms of $f_i$ with $\deg_t \geq n$. Then is there a sufficiently large integer $N>0$ such that the ideal $J_1 = (f_{1N}, \cdots, f_{rN}) \subset A[y_1, \cdots, y_n]$ defines an irreducible scheme?

(3) Under the above assumptions and notations, is there a sufficiently large integer $N>0$ such that the ideal $J_2= (f_{1N}, \cdots, f_{rN}) \subset \widehat{A}[y_1, \cdots, y_n]$ defines an irreducible scheme?