Obviously while transmission warms up!
Once i did a ls2/4l65e conversion. The transmission wasnt getting the corrects signals though. The transmission acted liked you describe. It turns out because of missing signals it engaged several clutch packs at once. A mile later the clutches were toast.
Try draining your oil and look/smell for clutch deposits while you change the converter.
With all this being said, i believe it is your converter though! If the transmission was shifting fine and dandy from 1st to 2nd until you broke it loose with the tires, the odds favor it is the converter..