Ok, I tried what you guys suggested...
And then some...
Not w/ the strat, not yet, but w/ the Les Paul.
I'm keeping a soft cloth at hand and wiping the strings and hardware at more or less regular interval, in between takes, and hopefully absorbing the excess of sweat on my fingers at the same time. Would probably seem obsessive was anyone is observing me, but, what do I care...
I also try to wash my hands during the session whenever I need to get up and stretch my legs a bit or need a break.
And I think it works. I've changed strings Monday night and they're still feeling surprisingly fresh, even daily 4-6 hours sessions. Usually after 5 days of such intense playing, they'd probably be in their last agonizing mile.
The other good news is that I managed to do the little maintenance job myself and won't need to pay someone to do it. As an extra bonus, I even managed to lower the action a bit. I always had difficulties adjusting to Les Paul's and this one was an exception, and it just went an extra step ahead in playability. It's now pretty much as smooth as my trusty old strat. Best of both worlds.
So the next step is to try new strings. I'm probably going to order a few sets of Ernie Ball Cobalt Hybrid Slinky. A bit more expensive than the regular ones, but I'm anxious to try them.
TCB - Tea, Cats, Books...