Interesting. 3rd uses both clutch packs, but the other forward gears only use the rear clutch pack, so my initial thought was a problem with the forward pack. But Reverse also uses the forward pack, so it doesn't seem like it's a clutch problem. Reverse also requires more pressure than the forward gears so it doesn't feel like a pressure problem.
Without thinking too hard it's either a hydraulic circuit problem (valve body or servos) or maybe a problem with the governor. The FSM troubleshooting doesn't specifically mention your issue.. The fact that you had water in it is definitely worrisome. Just draining the fluid won't get all the water/milkshake out. There will be some left behind in the torque converter at least. It doesn't immediately feel like a friction problem, but water does break down the glue that holds the friction material to the backing plates/bands so you're already dancing with the devil.
How high have you got the RPMs in 2nd while you're trying to make it go into 3rd? Let's establish if it's just missing or if it needs more RPM (i wouldn't think twice about winding it up to 6k RPM to test).
Have you tried shifting through the gears manually, M1 -> M2 -> D ?
Do you have any trouble with a kickdown shift from 2 in to 1? Or with it downshifting to first when you come to a stop/slow down? I'm wondering if the kickdown servo is having trouble un-applying or something.
The FSM does detail a series of pressure tests you can do with the transmission in the vehicle, although Reverse & D3 tend to "troubleshoot together" due to their common usage of the forward drum.
