A drill press doesn't usually turn fast enough or usually run long enough to destroy the bearings. What you are most likely experiencing is ancient, oxidized grease glopping up inside your bearings. ("Glopping" is a highly technical term we engineers sometime use.)
In your shoes, I would first disassemble and clean the old bearings with a good solvent. Be prepared for some rock-hard grease that has to be chipped off. Then re-pack the bearings with synthetic grease. (Any kind you can find will work because the duty cycle is so low on these machines.) Synthetic grease won't usually oxidize over time, so this is a lifetime fix. You probably won't have to buy anything but the grease.
I see this all the time while working on vintage motorcycles.