1987 Toyota Celica RPM's racing even at slower speeds

Transmission problem
1987 Toyota Celica Manual

My '87 Celica, manual transmission, sat in my driveway for a year and now I need to pass a CA. Smog check so put in a new battery, started it up, drove it around for about 50 miles (I need to empty the gas tank of old gas, put in new gas to pass smog check) with no problems, no leaks, no funny noises, everything was great.

This last time I started it up and within three blocks of my house, the engine (transmission?) Was racing the RPM's in every gear as if I was trying to drive in 1st gear at 40 or 50 mph. Every gear acted like that.

The problem seemed worse immediately after starting up from a stop at a red light and as I shifted to higher gears but when going downhill, it worked fine in 4th gear and stayed within the 2 to 3 on the RPM's.

There was also an odd almost burning smell but not like a fire or burning brakes or even a burning clutch. It was more of a metallic smell and it got worse as the engine strained under the RPM's.

There is no problem using the clutch and there is no problem or resistance or bad noises when shifting into the different gears. But, it's almost as if the gears are saying, "This is too hard for me. Don't make me work so hard." And I'm really not giving it more gas to go faster then not shifting to higher gear. The car just acts like I am doing that.

Could I just need new fluids or does this sound like a total transmission problem? If it's a big expensive problem, I'll just get rid of the car.