As it turns out, the cleaning was significantly more difficult than the liquid sandpaper. That was due to the horrid tape that the previous owners used to hold the drawers shut when moving it. There was still a lot of residue left even though I had worked on it for a significant amount of time last week.
This is what I broke out to attack the goo one last time:
The dryer helped to soften up the adhesive, and I suppose it's obvious how the blade helped... It didn't end up perfect, but it was significantly better when I was done.
Next it was time for my Liquid Sander Deglosser! My other tools were: (1) rubber gloves, and (2) a holey sock.
It is so, so easy, and I have read good reviews on it. It took me about 5 minutes to coat the whole dresser and all of the drawers. In the photo you can just barely see that there is a little bit of a film left on the wood. It should be just enough to help my bonding primer to stick. More to come next week once I stock up on supplies!