Is my db too big for my server? Should I go nosql or stick with relational db management?
I would like to echo @Thronk 's remarks about not having enough information to fully answer your question.

However, take a look at this article by a guy who sold his business to eBay and is still senior there.

This suggests to me that you would be far better off redesigning/refactoring your database according to proper relational principals rather than jumping on the NoSQL bandwagon which will involve more work and greater risk. As @Thronk points out, 22GB is not a large system. 

If you're using 5.6, try using the performance_schema (p_s) to pick the low-hanging fruit first, which may give you the breathing space you need to properly redesign. 

Don't be afraid to come back here with design questions and problem SQL - don't forget that the more detail you put into your question, the better your answers will be. It may be that some reindexing may help - again, don't hesitate to ask here. p.s. welcome to the forum!