To the best of my recollection, all of the budgets in the timeframe I mention were signed by the sitting president. (I realize this is not necessary for laws to take effect, but in the years I cited I believe the presidents all signed the budgets into law.)
If you want to argue that presidents are not responsible for laws bearing their signatures, I stridently disagree.
Presidents can only sign what is put in front of them. They have the option to veto, and sometimes do, but in many cases they may view that as the worst of two bad choices.
This position handily relieves the Executive branch of responsibility for any laws they sign during their tenure.
As Presidents typically take credit for their legislative wins, we also give them credit for budgets they sign. (And then we name them things like "Reaganomics.")
If you want to argue that presidents are not responsible for laws bearing their signatures, I stridently disagree.