I had the same problem on my single 1JZ. I also had crappy rusty coolant, and was losing just a bit of coolant (I think it was mostly getting pushed out of the overflow).
I did a couple full flushes and it didn't help. Still had the same cruising overheating (on my 45 minute drive to work, would only happen when I hit the long 15 minute cruise at 60+mph, cool as a cucumber from light to light, which usually means your fan is working decently, right?) and when heavily boosting for a long period of time (a few long runs in a row).
Eventually I just went through the whole thing because nothing short of replacing parts was going to fix it. It seemed like a small headgasket leak to me with the coolant overflowing a bit and overheating when driving and not idling.
Did the headgasket, put in a 2JZ waterpump, new radiator (had a cleaned up stock rad before), ARP head studs, ton of new seals, and flushed out everything like crazy and cleaned the block passages of a ton of rust and crap. The original headgasket was pretty rough, but didn't seem to have any obvious trouble spots on it. I think it was just a small headgasket leak.
So my advice to you would be to do the same. A 1J can blow a headgasket too after so many years and some single turbo abuse. I honestly think my hydro fan was still working well, it was pushing some decent air, but I wanted to put a new waterpump in there while I had everything torn down. That's too many years on an original waterpump, especially on a JDM import motor.