After you've swapped the preamp tubes over (V4 does Reverb) to rule out a bad tube, you can try taking the reverb tank apart after you take it out of the amp to check if it's corroded or there's any gunk inside.
Won't be caused by old filter caps but good to replace these if they've never been done.
Another thing is the reverb is pretty weak anyway on the III once you've gone back from another amp with strong reverb, but I'm guessing it was never like this before?
EV bee
Supposedly a Mk V tank should work though to save you taking the old tank apart to diagnose? Worth double checking with Boogie first though.
https://store.mesaboogie.com/products/reverb-tank-standard-tank-for-tube-reverb-circuits.html
Let us know how you get on!
Cheers,
Jon