If I trade in a phone I haven’t fully paid for in order to get a cheaper phone will I have to pay it all off before paying for the new phone?

You will need to pay for the old iPhone while you pay for the new one.

Buying a new iPhone and using the old as a trade in to reduce the price of the new one, does not change the terms of payment of the old one in any way.

The answer to that is subject to the terms of your particular agreement, but generally the entity who “fronted” the full price of the phone will require FULL payment as originally-agreed.