Because you have one award and one paid ticket (if I understand correctly), there are two different sets of "options".
For the award ticket, you can change to any valid routing between LHR and PUJ that has award space.
You can change the destination from PUJ to something else (again, with valid routing and award space). Normally this will incur a fee, but since it is due to a schedule change the "redeposit" fee should be waived.
For the paid ticket, changes are governed by the fare rules. An "N" fare from LHR-PUJ currently says that a change is 100 GBP + fare difference, with the 100 GBP waived for schedule change or death. If you buy a cheap ticket, it will come with the most restrictive rules regarding changes - that's a risk you took.
If you can find a route which gets you into Miami by 11:00, then AA might agree to it, but if you simply prefer to spend the night in city X rather than Miami, then they certainly aren't obligated to accommodate that.
In both cases, you have the option to cancel without penalty, which may be the only "common" option between the two sets.