For a flushing solvent, I just use brake cleaner. It will cut through the sludge and gunk and leave no luricants or anything behind. Do not forget to clean the magnet in there if you find it - it will be under a pile of fuzz.
(Or at least, there was one in the pan, retained by a clip when I did my Dodge. Will be doing my Chevy here before winter sets in)
Set the pan aside to drip dry someleace safe, face down - do not want to collect lint or anything else that is airborne in your garage's air. Brake solvent luckily air dries very quickly!
Now as for lint free cloths, goto Autozone and pick up a roll of "Finishing cloth". They are used for polishing vehicle's exterior painted surfaces, as TrailBlazin96 noted. Unwrap them from the adhesive label they come bundled in, they are like $2.50 for a pack of 5. Shake them out good, and wipe down stuff you got fingerprints or fragments of old gasket/sealant sticking around after your clean up from removal of the old stuff.
When you are done, you may wash them, but drip-dry them only! Oils left in the fabric may ignite in the dryer and cause you some trouble. You may re-use them if you wish, but I always bought a fresh pack of them to wipedown tranny surfaces. The $2.50 is worth it, IMO.