The 96-98 Cobras also use a T-45, but you don't want it because the crossmember mounts in a different location than the GT. Your best bet is to get a rebuilt T-45 from The T45 Source - Proven, reliable T45 Transmissions
Look them up, call and talk to Zak and he'll go over your options with you.
Although if the car does not stall in 1st gear that is probably the trans if you can feel the clutch grabbing unless something with the clutch physically broke. You'd have to pull the trans and get in there and see wth is up regardless and at that point you might as well get a new flywheel/clutch anyway while the trans is out. Upgrade to an 11" flywheel and clutch setup too if you get a new clutch. There is nothing else you need to do besides get an 11" flywheel and 01-04 11" clutch to do the upgrade there.
96-98 transmissions are notorious for going out anyway. Bad 5/R fork and the blocker fall apart after awhile. In 99 the were fixed with a re-designed fork and carbon fiber blocker rings and are much better.