Flush and relube first and see if that helps. Otherwise, Shimano road shifters aren't really intended to be serviced and most mechanics won't try (I won't, either). If everything else is in OK shape, I'd just buy some used shifters that had relatively light cosmetic wear--they typically last a pretty long time. If only one shifter is gone and the other seems OK, I'd probably source an exact used replacement to keep costs down.
Unless most of the parts are pretty significantly worn I wouldn't bother trying to upgrade to 11 speed--I'm all for 11 (and 12!) speeds, but an extra cog isn't magic and isn't worth the replacement cost if most parts are OK. If you do decide to upgrade, I'd probably recommend 5800 or R7000 (105 level). You can keep your current wheels if you use an 11-34 cassette.
Microshift shifters are honestly fairly nice but they work differently than Shimano--they have two small shift levers, a little bit like Campagnolo.