Yes, the air leak will definitely not be helped by not plugging up the oil inlet on the manfild, albeit a tiny leak.
Yes the '88 DTR ran the TMS26 carb not the TM28 carb found on all other years.
That said, I am not aware of any other changes. So finding a suitable inlet manifold (probably can still be had new) and the airboot shouldn't be hard. But finding a genuine Mikuni 28mm is trickier.
A split in the carb boot is not ideal, but it won't be the source of your problem. I'd just smear some silicone onto it and allow that to harden.
Crank seals are common causes for air leaks, as well as the gasket on the inlet manifolds (once you sort the oil connector out).
Spray some WD40 around the general vicinity and if the revs drop back to normal then you've found your culprit .