Should I use a two-prong adapter on a three-prong plug?

No, do not use a two-prong adapter. The third prong is the ground wire that prevents electric shock by giving stray current a safe path. Removing it with an adapter breaks that protection, and the adapter’s metal tab only works if the outlet box is grounded—which old two-prong outlets usually aren’t. Replace the outlet with a GFCI or use a grounded extension cord instead.