How do I makeover an old chest and dresser for a beach themed room?

That's a good project for chalk paint. Clean with TSP than paint and distress. You can do in a white or a soft aqua. Check out Rustoleum chalk paint and top coat. Very affordable, easy to use, can be mixed into most colors and top coat works great. Have fun.

Sherry...You can find some pretty napkins or wrapping paper with the beach theme on it then mod podge the dresser or chest. If you use the napkins just use one ply with the theme of course on it. Remove the pulls before you start the project. If you like you can paint some of it with coordinating colors from the napkins. Use the old pulls or find some new ones, such as shells, anchors, boats, lots of beachy ideas.

There is a really cool saltwash paint additive that gives a great beachy look. You can find it on Amazon here https://amzn.to/2UmAbOO You can also see how I used this on an old headboard here https://ourcraftymom.com/coastal-headboard-makeover-saltwash/

Others have covered the painting. I redid a large empire dresser years ago and a couple of details made it a beach/cottage piece.

1) I covered the unusable key holes by gluing on some clam type shells. This could be a accent even if there aren’t key holes.

2) I replaced handles and knobs with rope pullls. Might require drilling larger/additional holes. The ends can be knotted or stapled inside the drawers.

Napkins are a great idea, but I love Tatouage.com. Rubs on. Not a plastic sticker. Everyone thinks I painted it on. I would paint it to look like sand and water. The fish and ocean plants are realistic. You could blend the colors or make one the beach and one water. Have fun.

Some cute and unique ideas here:

https://www.completely-coastal.com/2016/01/dresseer-makeover-coastal-beach-nautical.html