You guess right it is running on the cold start injector only. Presuming you haven't been fiddling with things prior to this happening, check the earth in the top centre of the inlet manifold (brown wires from memory). Otherwise find your EFI main relay No.2, not sure where that is on an AE82 and check that. Am guessing it will be in the drivers footwell above the junction box as in an AE86. This switches on the injector circuit. Does it click when you turn on the ignition? It should as it has direct power from the ign switch. If not, problem is probably there. Swap the relay with another or have a look at the engine manual in the repository for the more detailed diagnostic path for the injector circuitry (pages FI 56 & FI 57)including how to check the relay. Good luck.