samsmix-
Your 721 probably has a two-piece sear (most of them did), made out of two pieces of stamped metal. The problem with that trigger design is that the sear wears and rounds off on the end, causing the sear to not hold when the bolt is closed or the safety is released. Sounds like this is exactly what happened to you. I hope no one was injured.
I know you said you don't want to hear this, but here goes. The easiest fix is to replace the trigger. Unless you know someone who is very experienced in firearms work and doesn't mind getting sued, no one will help you adjust or repair that trigger.
If you send the gun to Remington, they will replace the trigger. This usually costs abround $85 with shipping. You can call Remington for more information at 800-243-9700. Or call the Remington field service center for your area, which is Allison & Carey in Portland, OR at (503)256-5166. I would recommend calling Remington, as the field service center in OR is of no help in my experience.