Okay, so let me make sure that I have it straight.
The "AA" degree you're referring to is basically the general education requirements for a further degree (BSN in your case) and is not a terminal degree (in that it does not afford you the ability to sit for licensure of any kind).
The ASN does provide you with all the classes that you need to sit the NCLEX-RN exam upon graduation. From what I can tell based on the program requirements, the computer courses are addressed at the bottom of the nursing program page:
Students must fulfill the computer literacy general education requirements within the first 15 hours of BC credit by successfully completing the basic student technology test or pass the CGS1060C to earn the degree.
And the speech requirements a bit further up:
***Successful completion of the Nursing Program will satisfy the SACS oral communication competency standard
If I may reword your original question, I think that what you were really asking is if it makes more sense to do an ASN-RN program and begin working as an RN sooner (maybe do an RN->BSN program after) or to complete general education requirements (AA program) and then enter a generic BSN program. Do I have that right?
If it were me (and I've been there with a small child), I would go the ASN-RN route, get licensed and get working. As soon as you've graduated from the ASN program, turn around and apply to the RN->BSN program. You are out in the workforce sooner that way and your student loans (if you have them) will be deferred while you're in the BSN program (which was a big help for me). If you do the generic BSN program the way you have it described above, you will be earing an AA from Broward and then applying to a generic BSN program somewhere else. Why go for an AA (which nets you nothing in the short term) when you can go for the ASN and be a nurse much sooner? If you need to get into the workforce quickly, that's your best bet, IMHO. If you have time to take it slowly, don't need to be in the workforce quickly, and don't have to worry about childcare issues (aka not a single parent), then the slower route is fine, it'll just take you longer to become a licensed nurse.
(Oh, and I'm in Jacksonville, BTW.)