Yes, a 1.6 will fit I just finished installing one in a 96 Jetta, only one issue was p/s system, you need to use a 1.6 p/s and brackets as the blocks were slightly different bosses were the brackets mount.
Used a 1.6 non turbo engine and a 1.6 td downpipe mated to factory exhaust system behind cat. Kept all 1.9 cooling system/oil cooler on 1.6 to simplify wiring harness, used 1.9 clutch and flywheel. It was all pretty easy. Got tired of poor machine shop rebuilds (wobbling crank after machining TDI gear, low oil pressure) so client gave up on 1.9 idea.
It sounds like your engine overheated a bit too much, are the plastic coolant flanges all new? If so you might consider a different motor, most overheated diesels I have seen warped the head beyond spec and the cylinder bores were scored from swollen rings/pistons (not good), so cold starts were impossible. Usually the plastic melted is a good indication of too hot to bother rebuilding...
Look closely at your floors/rockers/ and underneath the rear end around bumper all soft spots on these cars, you might reconsider fixing the car is it is rusty underneath, I know most cars I see should not be fixed as your looking at a year or two before they fall apart (atleast here in Ottawa), very prone to rust out.
Also consider the costs of a new clutch, motor mounts, shifter linkages, trans oil seals, injectors, turbo oil lines (rusty?), p/s lines (rusty?), glow plugs, timing belt, oil seals, crank pulley, vibration damper, serpentine belt roller/tensioner/lever, is the injection pump starting to leak???
Lots of parts add up fast to the overall cost whether you rebuild or replace head gasket or find another engine.
Greg