If you really hate the company let them fire you and then sue their ass for discrimination. They have to provide you reasonable accommodations for your disability. A clean place for an exchange and light duty. You may need to look into this a bit further.
My hubby got LTD from his work , there was a waiting period , 2 weeks I think, for those weeks he used sick time, then LTD for the 5 months waiting period before the SSDI kicked in. I signed him up for both at the same time , as he was hospitalized on and off, mostly on, for 4 months . The LTD was a little higher than the SSDI , but they have been sending a check for that amount every month since. It's important to know that you have to not be working at all for 5 months before SSDI starts, and then it's anouther month before you get your first check . It was really nice to have that LTD check come every month to use through the long 6 months before a SSDI check arrives!!