My dryer will only spin if it is empty

If it's a belt drive, the belt could be stretched and loose, the tensioner spring could be weak or the motor pulley loose. I'd go for a belt or tensioner. Without weight the drum turns but slips when full. You'll need to pull the dryer out and remove the back to see what is happening. This is somewhat a generic answer since you didn't mention brand or model.

In addition to replacing the belt you will have to clean the drier out thoroughly. Every air duct will have to be removed and scraped and washed out to remove baked on lint deposits which is like a concrete lining. This will burn so the manufacturer has installed the safeties to keep from having a house fire. Driers were the number one cause of fires at one time. Look at: http://www.youtube.com/watch?v=MMP7BW1lLs4 to see how. Also: http://www.youtube.com/watch?v=iBiRw-KElC4 ; http://www.youtube.com/watch?v=yFTZGgVisCI
It is no use just to replace the belt because it will fail again quickly if it is not cleaned. When you disassemble it you will see how dirty it is.

Note that on most of the videos it does not show them washing out the parts, but this is important because it's the only way to really get the parts clean, just don't get motor wet or the capacitor if it has one. Vacuum out motor. I would move the drier out to your garage to have plenty of space to work. You also want to check your vent and clean it, also. The job is really easier than it sounds. There is no shortcut that will work; it has to be cleaned well. Before you re-assemble it make sure to use white lithium grease (high-temp grease) and grease all bearings. Just vacuuming out the dryer a little does little good. You also want to check the vent and clean it out. When you are done you will have a new dryer and it will dry like new again. You want to clean it like this every two years or every year with many children. DrTom4444