1992 Chevy Camaro Transmission wont shift to 3rd

I have a 92 camaro 305 with a 4L60 tranny all oem. All of a sudden it won't shift to 3rd, it just revs so hi. I don't know if will go to 4th either. I hear people say let it rev & it may go to 4th but that seems risky to the engine to me. My question is could it be a govenor, a vss, a clutch or band? How can I tell?

I sometimes have to drive it & in 2nd gear I can go about 35 mph before the engine starts to rev higher.
Will I mess up anything more by driving in 2nd for about 10 miles sometimes? Also the past few weeks the speedometer is intermittently working, but that just started, the shift issue has been going on for a few months.

The car ran great before this, so what steps shoud I take?