If I do SMP, would I stay in my unit, even if the branch I want to go into would be different?

In short it is up to you.
Do you plan on commissioning into the Reserve Component? If the answer is yes you will need a paragraph and line item from the new unit in order to be guaranteed that branch. Without a para/line number you will commission into the branch of the Army's choosing. The para/line number is what position you will be in when you commission. If that is the case switch units ASAP to give you more experience in that unit, or at least a unit of your branch type.
If you desire to go active duty your SMP unit of assignment doesn't affect your commission. Stay with a unit that teaches you how to be a 2LT.

You could probably stay in your unit. I don't believe you would be forced to transfer. After you receive your branch upon being commissioned you may then desire to leave your unit depending on what officer vacancies are available in your unit.

If I'm tracking the question correctly, you could very well stay within your current unit while PART of the SMP program....but you will certainly have to change units upon commission if the AOC you branch into is not slotted within your current unit.