I'm not sure I understand your question, but, from the title of your post "Mailing addresser", it sounds like all you're looking for is a machine that will print the name, address, city, state,zip, bar code, etc. in black & white on the addressee side of a pre-printed post card. Is that correct?
If so, the question becomes "in what volume and, at what speed"?
If you're talking runs of very small volumes (200 - 500) it can be done with any desk top monochrome laser, or, even a desk top ink jet (Rena, Buskro, etc.). If you're talking large volumes with high speed, you'll need something that has an "ink jet base" with a high speed feeder and an output conveyor (Kirk-Rudy, MCS, Cheshire, Buskro, VideoJet, etc.)
As to what type of surfaces it will print on and whether or not you'll need dryers, that is a function of the ink type you will be using on the ink jet. Ink Jet ink manufacturers (Collins, HP, etc.) have a "fast-drying" version of their inks specifically for use on low to medium aqueous surfaces. If it is "high gloss" (high aqueous, laminate, varnish, etc.) you will need a dryer on your ink jet as well as the fast drying ink.