i thought this wsa going to be some crazy story where the united states invades prussia and takes out poland and denmark. silly me :P

to your question: the a.i. almost never garrisons non-capital cities, so it may not be that they can't build units, but that they dont think they need to.

as to the trade: the only ports that count are in your capital and (i think) directly adjacent provinces. so even if they had 50 ports, if none of them were in or next to carolinas, they still cant trade.

what do i reccomend: either

A) take a province next to carolina on the coast, build a trading port, then give it back for a ceasefire,

or

B) trade a coastal province next to carolina with 1 you don't need, build a trade port, then swap it back out. if they wont give you back your province, at least give them theirs. for this reason, use your poorest region.