My guess is that you don't have real cheese in your sandwich.
I've seen this before with things called cheese that were really types of American cheese. It happened when then product was exposed to moisture that it seemed to absorb, which then caused it to turn soggy and glue-like. If you'd have said that you had tomato in your sandwich, I'd have been sure this was what was going on.
Saltpeter is potassium nitrate, which does not directly cure meats. Bacteria convert nitrate into nitrite, which is the real preservative. Saltpeter can be replaced by a smaller amount of nitrite to get the same curing effect (most commercial cured meats do this), though a prolonged cure that converts nitrate into nitrite can develop more flavor.
Tender Quick is not a direct substitute because it contains mostly salt. I've heard that you can replace the salt in your recipe with Tender Quick, and drop the saltpeter, and have a success. You would have better luck finding a recipe that was meant to use Tender Quick, though.
It is definitely possible to buy (food-grade!) saltpeter. I would check online, or at specialty stores. It's a little more difficult than picking it up at your local grocery store, of course.
(Chemistry lesson, courtesy of McGee: nitrate (NO3) is converted to nitrite (NO2), which then reacts to form nitric oxide (NO), which bonds to myoglobin in the meat, which turns it pink and prevents oxidation. Nitric oxide is also present in smoke, which gives that "pink ring" around the outside of smoked meats.)
Best Answer
In addition to the direction the corned beef was sliced, you need to boil the corned beef just below a hard simmer -- not on a hard boil. I have relatives who crank the knob to 11 and boil the thing into "stringy meat toothpicks", and that's no matter which way you cut it. I cook a 3 pound corned beef in 1 gallon of water on just a hard simmer for about 4 hours and get fantastic results.