Hi, Angelo. For a permanent fix, get some shellac, denatured alcohol and a foam brush. Be sure that the shellac you buy is less than 6 months old and the newer, the better. It will be dated on the can. You can buy the denatured alcohol in the drug department of a mass marketer or drug store cheaper than you can get it in the paint department of a home center, but, of course, you can get everything there at the same time.
In an old 1 pound coffee can or similar, mix 1 part shellac to 5 parts alcohol. Take out the drawers and coat the interior wood with the foam brush. The liquid will be very runny so be careful that you do not get any on the outside, finished parts of the drawers. Then coat the inside parts of the piece again being careful not to get the shellac on the outside. My guess is that there is some poplar in the wood of the case of the dresser and that does smell bad.
The denatured alcohol is very flammable and the fumes can get to you. Be sure to do the painting outside if at all possible and have plenty of ventilation for what you have to do inside. Stay away from all sources of ignition and the flammable fumes can travel. The alcohol will quickly dry/evaporate and should be ready for odorless use in a couple of hours or less.