I don't want to bite down or chip my teeth while I sleep. Should I try a mouthguard and are drugstore ones ok?

See your Dentist. Mouth guards work great, but --- not the drugstore ones which are not ok. They tend to fit poorly and therefore cause the wearer to clench and grind more! Often times, a dentist supplied guard is not much more costly and will fit precisely and last longer. See your dentist who will explain the various types and designs and custom make one for you.
Mouthguard YES! Drug store guard--no! I do not recommend the types sold otc--they lack durability and are ill fitting which can effect your occlusion (bite) in the long term. Visit your dentist and ask for a custom fitted night guard/day guard. Your dentist can help determine which type of guard will suite you best to protect your teeth while you sleep.