Hi. So I've been having this problem recently as I've been playing more and more often. I'm finding that some of the keypads, particularly on the boot joint, are sounding kind of sticky, and when I take my bassoon out to start to play, I find that some of the keypads stick to the bassoon a little bit. Luckily, it hasn't created a problem, because the keys that are sticking are one's that are lifted off of the bassoon (such as the A-flat key and the B-flat key). I was told that this happens because of moisture that gets under the keypads. However, I swab twice every single time I clean up my bassoon, and the keys still stick. I now will even take paper towels or pieces of paper, slide them under the keypads, and press down on the pads to dry them off, as I was told to do, and the keys are still sounding sticky!

I was wondering if this is a common problem, and if so, what I can do to stop it. I want to handle this issue before it becomes too problematic, such as causing keys to stick to the point that it affects my playing.

Sounds like you need to get the pads changed. Some types of pad do go this way.
Also it can be due to the varnish used on the tone holes being a bit sticky.
I'd suggest taking it to an expert bassoon repairer for advice. Renewing pads yourself is not a thing I'd recommend trying especially the big ones on the butt joint.

I did bring it to someone earlier to get one of the pads checked out, and he said that I'm just a "wet player" and that the wetness of the pads isn't something that needs to be brought in and fixed. Should I disregard this guy's opinion and take it in to someone else? Keep in mind that this guy may not be an expert.

I think moving to pad replacement might be a bit hasty, especially when someone experienced thinks they are all right. Regular swabbing is a must, of course, but I'm not sure about the paper you use to clean/ dry the pads. The idea is good but I certainly wouldn't use blotting paper as it is easy to leave a (microscopic) residue on the pads with any kind of "loose weave" paper. There is a paper specifically for pad cleaning which has a makeup similar to cigarette papers but is more robust. Mine is made by Yamaha but there are various makes which should be available in your local music shop. You use it exactly as you describe-- dragging it across the pad under pressure.
If you really are a "wet" player you may consider chocking open one or two offending keys while your bassoon is not being played -- low F sharp and the like. Just use a golf tee or similar.