Also,
there is a larger commercial airport nearby (still class E). Of course I
want to avoid that airport, but what's the best way? Should I just plot a
course around it or will they want to vector me?

(I should have added this to my previous reply.) I would just plot a
course around it. They may want to suggest headings to you, it would
probably be a good idea to cooperate (though it is not required). If
you have two radios, keeping one tuned to the CTAF of that airport as
you go by is probably a good idea.

You can also fly above pattern altitude, say, 1000 feet above TPA.

Jose
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