i do the "cheek test" with the back of my hand. if it feels cold, there's still moisture inside. If you are force-drying, there is always the risk of cracking and is usually due to either too rapid or uneven drying. airflow (or better - warm air) is the best for drying out work as opposed to straight heat -- temperature, airflow, and surface area all effect rate of evaporation.
ultimately, yes, you could definitely put a wet piece of clay in the kiln and fire it without issue - the problem is that you will have to adapt firing schedule to accommodate the piece(s), and have a longer hold/preheat. most likely with the firing schedule of your studio and peers' work, the schedule used doesn't work out in your favor and you get broken pieces. In my studio, we typically once-fire everything (all sculpture). Because students like to wait until the last minute to finish or glaze work. It is not uncommon on kiln loading days to encounter work that's questionably past leather-hard, or literally just got a quart of glaze soaked into it, lol. Luckily, I fire conservatively enough to seldom blow up a student's work. I know, we're crazy and like to rock the boat with our technique, but it works
like Neil said above, your clay will ALWAYS have water in it until you fire the clay. Even if you throw a tiny little shot glass and let it dry for a year, you will still have water inside due to the moisture in the air. If you dry your clay out in an environment with ZERO water in the air, you will STILL have water in your clay that could potentially make the piece explode....this is the CHEMICAL water that's molecularly attached to your clay particles. This water is present until the clay gets fired past something like 500*F, and is a permanent change. If you have a very thick piece of clay being fired at too fast of a climbing rate, the piece can and may explode during this window where the chemical water is being driven off.
Want to do a test? Everyone always claims you can't put wet work in a kiln without it blowing up, if you go past boiling point (212*F) because the water turns to steam and blah blah blah we all know this explanation. Try taking a piece of fresh clay (like say a 3"x12" solid pug stick) and put it directly into a kiln holding at 300-350*F, well above the boiling point of water. Come back after a few hours to look for exploded bits of clay everywhere and you likely will NOT find it. Sure, the piece may crack like crazy to let moisture out, but it won't explode the way you think it would. It's actually quite fascinating IMO and might be very helpful when it comes to firing certain items.