So I researched the repair options for sticking Shimano brifters, and the advice I came across in various forums said to douse them in kerosene or spray with WD-40. I chose the latter, and after spraying about 1/4 of the large bottle in them, am happy to report back it's shifting great, all except the 10th and final speed. I oriented the derailer so that it pulls up to the largest cog, but doesn't reach all the way to the smallest 12T cog.
Thanks interwebs!!!

Here's a good video showing the process, although I didn't take them off the bike: