A bit late to this but we have been struck by this too. From what I can see it's likely a moth rather than sawfly. There's a couple of things suggested online:
Keeping the tree 'open' will help birds and bats move through to eat up the offenders.
Using pheromone traps that attract the male moths (only) and stick them to the paper.
Removing/burning fallen fruit.
The lifecycle seems to be that the adult moth lays the egg on the plum and it falls to the ground. There on the ground the larvae overwinter before emerging next spring as moths and starting again.
That leads me to believe that having chickens grazing under the trees will help, though I don't see much about it online.
The sticky resin isn't a sign of a rotten plum but something else entirely (I can't remember what just now). I've tried guessing which plums are infected but I can't always tell. One thing that we read, and did find true, was that the infected plums ripen earlier. At the start of the season a lot of the plums we pulled off the tree were infected and a whole load of windfall plums (maybe 1 or 2 in 10), but later in the season and it was very rare to find an infected plum (maybe 1 or 2 in 100).
We've pruned our trees and done a good job of clearing all the windfall plums so we are hoping for a few less worms next year. Need to get the chickens under the trees though.
Dans