When was the last time you optimized your tables?
What size is your DB? What version of PHP? Is this issue 24/7 or only during peak times?
If your DB is large, you may have outgrown the resources of what your shared hosting provides.
For example, one of my larger UBBT sites DB is about 1.5GB. Its hosted through Bluehost on their "Professional" shared hosting plans. But if I were to switch my large forums to their entry-level/basic plan, I would be having the same issues you describe. Why? Because entry-level hosting packages tend to pack more accounts per server, which leads to less resources per account.
For example:
BLUEHOST shared hostingBASIC (Standard Performance) vs PRO (fewer accounts per server @ High Performance)
https://www.bluehost.com/hosting/sharedSTABLEHOST shared hostingSTARTER (500 accts per sever @ 1 Full CPU Core) vs PLATINUM (50 accts per sever @ 2 Full CPU Core)
https://www.stablehost.com/web-hosting.php
