Yes the TRD-12VDC-FB-CL relay look suitable. The pin layout looks the same as the original and the coil resistance is the same (400 ohms.)
I think the dremel is a good way to open the old relay. Maybe you could post pictures of the faulty relay when you have removed the case.

Les.

 

Thanks for everyone's help. I should get the diodes tomorrow and be able to put this all back together. I'll try to open the relay later today (may be tomorrow) and take some photos.

 

Well, the contacts were welded together. Hope these photos show well. This is not how I envisioned the internals. I envisioned a contact on the right and one on the left with a T crossmember. Not everything in the middle.

 

Thanks. Yes I replaced both with the TRD-12VDC-FB-CL relays that I had on hand. I've plugged the unit into 120v and the fuse didn't blow. Haven't connected it back to the chair yet but I believe it is as good as new.