Some great advice above, nothing I would disagree with. Some extra points of my own:

- You mentioned potentially changing your flights. In a way you have got them the wrong way around: you should ideally swap airlines around, and arrive into London on the BA service from Sky Harbor - since that has moved to T5 - and return on AA, departing from T3, this would avoid the terminal transfer in both directions.
- If you are the sort that hates being rushed between flights, well yes, try and increase the connection time. The downside is that it makes the overall journey longer / more tiring, so I would personally avoid that, but maybe a slower approach is better for you?
- You didn't have a great transfer last time, but you seem to have lounge access. T3 has some great lounges for you to enjoy (some would say the best selection of lounges anywhere) and the lounges in T5 are better than what you will get in most airports. So maybe consider leaving a day early, having a night in Heathrow in both directions? The hotels near the airport are very good value on the whole, lots to choose from and will mean you get to BCN or PHX more rested than the direct services. The security experience for those starting in London is generally better than the experience of those connecting, for complex reasons, so that will de-stress the process too. If you did that the T3 to T5 thing wouldn't matter. You could choose if the luggage goes all the way through to FCO out / PHX back, or you can recover in London, assuming it's one ticket. That also protects your cruise timings a bit better, if there were delays.