The best way to get better at the visualization part of calculation is to practice it -- so yes, not using the analysis board anymore would be a good thing to do. In my (limited) games on this site I've been analysing the board in my head as if it were a "real" OTB game and only after getting a very good feel or even deciding a move will I check it on the analysis board. If there's some really complicated or long tactics I might pull it out sooner.
As for switching from 2d to a 3d board when you played your cousin, I think that's normal. I also used to only play online, and when I tried to play on a board I had trouble, but after some practice you'll be able to do both without that handicap. And if you switch and only did 3d boards after a while you'd have the same problem but reversed :)
It seems to me that how you perceive the board is an extremely important aspect of the game. You could be a great player, but without being able to imagine moves ahead there's not that much that can be done.
I have played chess on an actual board a few times a while ago, and I was pretty fine with it. But I recently took up chess again, but this time on the computer. For me the Chess.com board is really nice, and I could really see the game using this board. Today, I played with my younger cousin who had a board, and I realized how much I couldn't really see with the actual board. It was as though I was playing on a completely different battlefield.
Another thing is, on online chess, there is an analysis board. This allows me to see the actual move combinations which makes it very easy. Otherwise, it's really hard for me to see out variations, especially longer ones.
I don't know if I'm ever going to play in tournaments or on the board a lot more, but I feel like it's something I want to be able to do, if I'm playing chess.
So how can I improve seeing an actual board, but more importantly imagining the pieces and being able to foretell various future moves and variations in my head? I think for one, I should try to stop using the analysis board on the Online chess. I would think something like blindfold chess would be sort of helpful (but this seems to be more on remembering the board, but also I don't got anyone to play that way with).