A good grammar checker should pick up various mistakes a spellchecker doesn’t, including subject-verb agreement and using the wrong homonym. However, this is no guarantee, especially if you tend to write complex sentences.
Homonyms are seldom if ever picked up otherwise, so it’s best if you memorize them. The following meme is handy for their, there, and they’re.
One homonym I’ve seen used incorrectly numerous times by different authors is shutter instead of shudder. A shutter is a panel to cover a window; shudder is what you do when you’re scared and can be either a noun or a verb. There are various lists of the most common ones available if you google “homonyms.”