Is there a way to check what the SofL is for various offenses? I assume for non-felonies it’s state to state. Is there a way to look up various crimes by state to find how long they have to catch you? Also, is there a way to check for open warrants by state short of subscribing to some service or actually calling a jurisdiction?
Please just give me the (heh) legal ways of finding this out. 
Yes. It’s state by state for all claims/offenses, whether felonies or not, unless it’s a Federal (statutory) claim or offense, in which case it’s Federal. Unsure, not my area.
(It’s possible that you’re assuming that all felonies are Federal. That’s not the case. Most are state.)
The limitations are usually in the same chapter of the state statutes as the crimes themselves. Find that chapter. Usually a state will gather all its criminal statutes in one chapter, known as the criminal code.
The statute of limitations for Washington State is here.