I have had same cases in my shop. It is very likely the carb problem. I also have a craftsman LT1000, 12 years old I believe. If you do not want to tackle the carb removal, there is a trick you can do:
1- Use a clamp or small flat nose vice grip pliers to stop gas from going into the carb bowl.
2- Start the mower and use all the gas in the carb bowl.
3- Go get a spray can of carb/choke cleaner
4- Pull the gas line off the carb where it enters the bowl.
5- Fill the bowl with the cleaner, make sure it is FULL
6- Let is sit in there for about 20 minutes.
7- Hook the line back up to the carb
8- Remove the gas line blocks, fire it up.
If this does not do the trick the first time, repeat the steps again.
Also, there is an anti-backfire solenoid that screws into the bottom of the carb and has a wire attached to it. Have some sit on the seat and push the brake pedal as if ready to start it. Take an 8" or so screwdriver and put the tip against the bowl, put the handle against your ear as if a stethoscope. Have someone turn the key on-off several times. You should hear a 'click' when the key is turned on-off. If not, it is likely the solenoid is either gummed up and not opening enough to allow gas to flow through the main jet or not working well. It takes a 1/2" wrench that has been ground thin enough to reach the 'flats' between the carb bowl and the solenoid, Just unscrew it, then take a pair of cutters and cut the plunger off. Screw the unit back in. Besides, the solenoid is pretty much useless, and IF the mower does backfire, so what..... it means you have just turned off the mower and a little 'pop' of the backfire is in celebration of your just finishing mowing the yard!