Couldn't wait for me to get a chance to speak for myself eh?
It is pretty clear that WyoDOT and Sweetwater County did a maintenance swap but haven't bothered changing any of the route signs. It is worth noting that WY 370 isn't signed terribly thoroughly to begin with - BGSes from I-80 do not make any mention of it, for one. This is as far as I could spy every sign there is.
At the west end, heading eastbound away from I-80:
At the former east end, heading westbound away from the general aviation parking lot:
And northbound on former CR 45 / the current paper WY 370 alignment:
(Former) CR 45 is signed with a single reassurance pentagon at each end.
As for the what to do on this....
This makes it a bit tricky. Isn't the precedent here to remove the part that we know has been decommissioned, even if it is still "signed" as a state route (if only because shields haven't been removed)?
Ehhh.... not really. Yes, precedent is to go ahead and truncate officially truncated routes even if signage has not caught up. But this isn't a truncation, it's a reroute. With reroutes, precedent is generally to leave it alone until signs are changed.
I do suspect the signs will be altered to match the new routing eventually, and WyoDOT just hasn't gotten around to it yet since it's not a high priority for them.