When you have to run your engine with the choke close that means that you are getting too little fuel. So when you close the choke shutter, you have less air and more fuel,(albeit a richer mixture) and your engine will run.
You can do two things, first right under the carb is the float bowl and on the bottom of it is a brass looking nut with a screw going through the center of it. This is your high speed adjusting screw. Run your engine
for about 2 minutes to warm it up then put it to full throttle and turn that screw maybe 1/8 to 1/4 counter clockwise to richen up the mixture and see if your engine will now run with out the choke off.
Second, I would say that it's probably time to give your carb a good cleaning which would mean complete disassembly, soaking it in an approved carb cleaner and installing a carb kit. Todays gas goes bad so quickly that if you have gas older than 30 days old, I recommend getting rid of and getting fresh stuff.
When you decide to tackle the carb job, so to your local small engine dealer and purchase the Tecumseh Mechanics manual. It's inexpensive and an extremely good read. I highly recommend this book to anyone who wants to learn more about their engines. I believe the part number of the book is 692509.