On a full house, then river card is second ace

My question is how I should have reacted to the appearance of the ace on the river.

I think either checking and calling a small bet or making a small block bet would be fine. Since it is quite likely he has an ace I don't think it is wise to bet big there. And if he doesn't have an ace he is not going to call. So by betting big you just give more money to villains good hands and do not get any value from his weak ones.

Also, was my opponent playing right? I feel he should have put me on either a set or at least top two-pair, which would've been beating him till the river, so his calls seem like bad play that got lucky.

I haven't played much live poker but the dynamics are different there. The board was quite dry so you rep a stronger range but I think your double barrel doesn't necessarily always mean you are very strong here. I don't think your villain made a huge mistake by calling OTT.

Thanks. That's really useful. I like the idea of a small bet there and I'll probably do that next time in a similar situation. I would like checking and calling less, in case he bets big thinking that I am weak.

I definitely think he was right to call my flop bet, as I might have been making a bluff contibet when the ace came on the flop, and he has top pair. It's just the turn bet that I don't really agree with.

My logic is this:
I have raised, so presumably have a reasonably good hand. I have now bet again on the turn, so probably didn't have something like KK, QQ or JJ. Maybe, therefore, I have 10-10 (and hit my full house) or Ace-high card in the hole or even AA. With AA, he cannot win. If it's Ace-high card, then I have him beat with AK, AQ or AJ, as my pairs of Aces and Twos beat his. A-10 beats him with top two-pair. He can probably discount me having less than his A9, as why would I raise unless I am loose? So my analysis is that he should realise he's virtually certain to be behind and has only two (if I hold an ace) or three outs on the river, which makes calling a big bet rather poor.

But I am pretty inexperienced, so maybe I am wrong!

From which position you opened?
What was your table image? Did you bet aggressively often?
Did your villain open a flush draw OTT?

I did think about the flush draw... That's why I placed a big bet, so that he would not have good pot odds to call me. Perhaps he thought it was worth it.

I raised from MP1, I think, and my image was definitely TAG at that stage. I had entered few hands.

Thanks for the advice, anyway!

In case he opened a flush draw OTT I think it was a decent call from him, since he had nut FD+top pair and in some cases any A or 9 would be his outs too.

I think A is pretty bad river card and I would c/f. I think we can call some small bet, but I think most players will not try to bluff you out.