The ASM loaded mode test used in CA measures HC in PPM so your numbers don't make sense. Now if you're refering to CO as 5.4% you indeed have problems. Post the actual numbers when you can. As Grim said, an obvious issue must have been found in your EVAP system for it to fail the visual.
High HC can be caused by many things including lean misfire, carbon deposits, even a faulty EVAP system. High NOx can be caused by faulty EGR operation (the EGR system is there solely to control NOx), high combustion temps, a lean mixture, faulty cooling system, fan clutch, lots of things. The tricky part about NOx is that it's maximum occurs very near stoichio so it doesn't take much of a lean condition to drive it up.
Throwing parts at the car isn't the way to go. Yeah, a new converter will likely clean up the mess but if the root cause is still there it'll just reduce the new converter's efficiency and shorten it's life. Better to check things out first. Any codes? Is the O2 sensor cross counting? Check it at the diag connector. The sole job of the sensor is to precondition the exhaust stream to the narrow range a cat requires to operate. That's why it's there, not to make the engine run well, because "well" is a relative term. Drift away from that ideal exhaust composition and not only do emissions go up because the mixture is wrong but also because the cat will no longer work at peak efficiency. If it's old than guess what happens? The bottom line is anyone who runs a cat must keep the EFI running perfectly. Those who ignore faulty engine management for anything more than a little time will be buying a new cat in addition to chasing the EFI problem.
Cali does indeed have tough standards but not so bad for 81-87 model years and many cars pass without EGR. My 1981 DMC 12 doesn't have EGR but it passes every time for NOx because it has a big cat. I also check the lambda system before I bring it in for testing. Which reminds me....
One of the most common mistakes made is to replace a factory cat with a cheaper one. Cheap means small because the cat uses precious metals. Small means low effciency so again, if you're going to skimp on the cat you'd better stay on top of engine management. Also, the Supra uses a three way catalyst so one portion could be dead while the rest is OK. Iin the case of NOx that would be the reduction bed. Is the cat lighting off? Does the engine run hot? Ignition OK?
As a formerly certified California BAR 97 emissions tech it's difficult for me to help beyond this without the previous test results. Post them when you can and anyone else who asks for emissions advice should always do the same. The relationship between the gases can be very helpful in diagnosing where the problem is, especially the O2 level as O2 passes through the cat without being effected. That said, the best way to diagnose emissions problems is to sample before the cat, assuming you have access to an analyzer.
Or you could just throw parts at it
