Hand 1: I guess Zeebo's really applies here. The board is double paired and while you hit it for the full house, it's the lower pair and someone might have the bigger one. My immediate thought here is that with $0.07 in the pot on the river your RELATIVE hand strength does not mandate you betting. You could just check/call. If you check the opponent might not have the idea that you have a hand that you can call a big bet with and just bet $0.20 or something which you can then call and get off the hook cheaply. As played when he bets all-in - he probably has either a J or a 4 (that is, the full house). Since there are two Jacks unaccounted for and only 1 4 you are most likely to lose. Your bet is big enough that he's not shoving all-in here as a bluff imo. I'm not saying I would find a fold here either, but it's probably the best decision.

Hand 2: Bet flop. Your turn bet is smallish and he might think you're just taking a stab at it. On the river I think he's shoving for value - but he might be doing that with a T thinking he's best. Your hand is underrepresented and I think his flop and turn checks are just a bit too weak for him to hold an ace a lot of the time. He could easily think you are holding a 4 or 5. I'd probably call here.