Well done on your achievement!
You've caught on very quickly! I used to teach kids who had been riding for months and were still not comfortable enough to be able to trot and steer unassisted.
As far as your query about using a whip, personally, I dislike novice riders to get their hands on crops or schooling whips - I have seen a lot of people become very reliant on them, when there's really very little need.
When you "sit" in the trot, apply pressure then, and release to rise. The whole movement becomes a sit-and-squeeze and keeps the impulsion going. I agree that kicking constantly causes your horse to become dead to the leg aid, but releasing the pressure and then re-applying should keep his mind on what you're asking for. If you're still sitting to the trot anyway rather than posting, then feel for the moment that the inside foreleg goes forward (when you would sit if you were posting), or ask your RI if you are struggling to get a feel for it, and push on then.
If your instructor does feel that you need a stick to keep the impulsion, use it only as a back up to the leg aid, e.g. if the horse doesn't listen to a squeeze, go for a firmer kick, and if there's still no response, then you tap with the crop behind your leg. Ask, tell, insist. I had quiet words with a young girl at the weekend who mounted her pony outside his stable and then cracked him over the backside as soon as she was in the saddle when all the poor thing had done was stand nicely for her to get on
. She claimed it was to "wake him up".
Good on you, and good luck for your next session!