There exist co-spectral trees. As there are no cycles to preserve, they fit the bill.

As far as 2-connected graphs are concerned, I would try finding a 2-connected graph $\Gamma$ which can be obtained from disjoint graphs $\Gamma_1$ and $\Gamma_2$ by identifying a pair of vertices $u_1,v_1\in V\Gamma_1$ with $u_2,v_2\in V\Gamma_2$, so that the Whitney twist, i.e. identification of $u_1,v_1$ with $v_2,u_2$, gives a non-isomorphic graph, but preserves the characteristic polynomial of the adjacency Laplacian matrix of $\Gamma$.

2 added a possible approach to 2-connected case

There exist co-spectral trees. As there are no cycles to preserve, they fit the bill.

As far as 2-connected graphs are concerned, I would try finding a 2-connected graph $\Gamma$ which can be obtained from disjoint graphs $\Gamma_1$ and $\Gamma_2$ by identifying a pair of vertices $u_1,v_1\in V\Gamma_1$ with $u_2,v_2\in V\Gamma_2$, so that the Whitney twist, i.e. identification of $u_1,v_1$ with $v_2,u_2$, gives a non-isomorphic graph, but preserves the characteristic polynomial of the adjacency matrix of $\Gamma$.

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There exist co-spectral trees. As there are no cycles to preserve, they fit the bill.