Andie, if you want it to last, you are working on the wrong material. The MDF (medium density fiberboard) will always be ready to absorb water and expand like a sponge. Later, it will dry and move back a little. This will not be much, but more than enough to destroy any glue joint. The laminate on the outside will keep it looking great until it falls apart one day. And I speak as someone who had a lap full of computer one day because of this kind of material. My current computer desk is solid oak or oak plywood and probably 60+ years old. And it does have a replacement piece of real linoleum on the desk top.
So what do you do now, Andie? My suggestion is to look in thrift stores for solid wood furniture and refinish that--or just clean it up. We live in a fairly nice suburb and I just found a store that obviously gets a lot of stuff from either "pickers", who drive around checking the stuff on the curb before trash pick-up day, or from the guys that work on the trash trucks. I couldn't believe the quality of some of the pieces that they had for sale and the reasonable prices. Since you now know what the bad (mdf) stuff looks and feels like, and how to take a drawer out to check, you might want to explore this option. It would probably be less costly than repairs on something that will still not me worth much.