Luckily the last thing you need right now is heat! But it is smart to check everything out.
That click sound was that from the furnace or thermostat? If from the furnace, most likely that would indicate power. Lots of spots that could be the issue. Maybe DOA control board. loose wire connection, bad fuse. I'm sure it is not the sail switch, nor the flame sensor. As the furnace will start up, The fan will run, and shut down 20-30 seconds later.
Personally, I would grab a multimeter, and do some real testing. I would start with the easy items then work towards the harder things. Check the fuse for continuity, visually it might look good, but still be blown. Check the fuse slot to verify power to the fuse (converter side of the slot). Check the power leads at the furnace to verify you have proper power to the unit. Confirm the thermostat wires are connected on both ends. You can cross the wires to simulate the thermostat is calling for heat. Check the ground wire from the furnace to the TT chassis, for continuity.
The hardest thing to check is the backside of the power distribution panel for loose wire terminal connections (load side of the fuse). With all the power disconnected, both shore power and the battery. Remove the entire panel, I think it is four screws, behind the bezel. The main shore power wire, will be wire clamped to the floor about 6" from the terminals. You have to remove this wire clamp before you can really get the power distribution panel out. On my TT, I could just get the panel out far enough to blindly reach in with a screw driver (#2 square drive) and loosen the wire clamp. Once it was loose, I could easily remove the panel and check the terminals. I found lots of loose terminal lugs. Make sure when you put the power distribution panel back, to secure the main 120 Volt power cable to the floor. I added a few additional inches to make accessing the next time easier.
Good Luck, but this is also why you have a warranty. but I would try to fix it myself.