Is your car's engine getting up to temp.
A faulty thermostat can play havoc with the tranny.
If it's an electroinc type tranny (ECT)whicj I'm pretty sure it would have, then the engine needs to be at certain temps for 3rd and O/D to engage.
I recently put a newer 5mge donk into my Cressida to replace the the old 5me. The newer engine has an electronic tranny and I had troubles getting the ECU to change into 3rd and 4th. I have since figured out that for 3rd gear and 4th to engage then water temp need to be a least 35degrees+ for 3rd and for forth gear the temp needs to be around 55 degress+. My thermostat was jammed open so it took forever for 4th to engage if at all. 3rd usually came into play after a minute or so after the engine was started from cold.
It can't be the tranny solenoids because for O/D both solenoids are off, unless one is jamming. The two solenoids are used for 1st,2nd and 3rd engagement, besides a fault code will appear if a main solenoid is faulty. The O/D cluster lamp should be the check lamp for the ECT ECU too.
I might be on the wrong track but it's a start. The temp sender could be dicky too, is the car hard to start from cold, ie does it take a few seconds to fire whilst cranking.