Playing capture and keep can address that, also, I think if you used scientists or tax collectors from captured towns. But, that's almost surely not going to work if you have multiple opponents when taking on Wang. If you can't keep a town, you can still sell off all the building in it before abandoning it. That's still better economically than razing.
Slaves aren't likely to be much of help economically. Likely playing capture and keep or capture, sell, and abandon is better economically.
Unfortunately, Korea had Artemis for such a long time now, that they acquired a big culture lead, so I'm afraid that by keeping the Korean cities, we might get into more trouble than it's worth, losing units in flips, having to divert more units to recapture flipped cities in our back (or even worse, on our supply line... if we have secured 1-2 of the Korean lux resources, and a flip disrupts the road network to those resources, our entire core would revolt for a turn, and we would have to raise the lux slider again for as long as it takes to restore the connection... I went through all this before...

)
So my advise in this situation, at least against Wand, would be to not try and keep any cities, unless it has a useful wonder.
The way slaves can help economically, is by reducing unit upkeep: there is still some work to do in the semi-corrupt area, we will need roads to the frontline or to important resources, so we still need a sizeable workforce for quite some time. But for every two slaves, we can join one of our native workers to a town/city and reduce the unit upkeep (plus increase the output of that city). So slaves are quite useful.
Selling and abandoning is also an option, but in my experience not as useful as the slaves from razing. (You don't get any slaves, if you abandon a city after capturing it.) How much do you gain by selling the improvements of a captured city? Cultural buildings are auto-destroyed upon capture. The aqueduct cannot be sold. And of the remaining buildings, about 50% get destroyed on average, when capturing the city.
Smaller or corrupt cities don't have much to begin with, and even if you are lucky and capture a big productive city that has built temple, library, cathedral, university, market, courthouse, granary and aqueduct (which is quite a lot...), then if you are very lucky, you can sell market, courthouse and granary for a total of 60g. If you are unlucky, all three of those get destroyed. Then you have nothing. And on average, you can expect 1-2 buildings for let's say 20-40g. A slave is worth approx. 110-120g (that's the price for which you can usually buy them from the AI). Razing a big city gives you 5 slaves or 550g. I would take these anytime, if I have to chose between them and a 50% chance for 60g cash...
(Especially, if this game indeed goes into "overtime" and we need them later for rails...)
The research rate would be quicker, yes. ONCE you start to research that tech. But, the overall time to acquire the technology will likely be longer. A 50 turn run and two 8 turn runs is 66 turns. Three 20 turn runs is faster than that in acquiring the technology.
That's true. But the benefit of all those extra towns (in unit upkeep and for producing either income or trebuchets or workers) is permanent. And it's not so easy to weigh this up against the benefit of having the techs a bit earlier. As I said, everything in Civ has its up- and downsides... Takes years of experience to get a feeling for which option might be "best" (whatever that means) in a given situation.
Unfortunately, you cannot look into a city before deciding whether to raze or to keep it... The slaves from razing are guaranteed, but everything else is a gamble...