It very well could be your transmission, especially given it’s 16+ years old.
The rubber clutch piston seals do harden over time. Hardened seals do not seal properly, eventually leading to slipping clutches, which can cause what you’re seeing.
There are additives that can be used as a band-aid to temporarily swell the seals. They do work, though not forever.
As others have noted, you may want to take it to an locally owned independent transmission repair facility.