How do you pronounce "a" in a sentence?

I say "a" (as in "bat").

It annoys people because it's a weird way to talk, so I keep doing it.

It will all come down to country/area/region, coupled with what type of education someone has had.

English is a stress-timed language. That means that the main words get a longer time to be pronounced than the short, functional words between them. That makes those words shortened and contracted.

"I gotta go to a barber."  I, go and barber are stressed. Everything else gets pushed together.

Unstressed vowels are usually pronounced as "uh"- that's the schwa someone mentioned. If the syllable or the word is stressed, the pronunciation changes.

"I said i wanted A (long a as in hate) potato, not 5 potatoes."

It doesn't matter much where we're from- Americans and Brits all do it this way. 

I was taught that it has its short sound in a sentence like “hat” as opposed to its long sound like “hate”. Which is quite the opposite for “I” in a sentence which uses its long sound like “kite” as opposed to its short sound like “kit”. 

Pretty much the schwa sound: "uh". But the long A (as in hay) is not incorrect. There is no english sound made with 'e' that would be correct.

It sounds like the "a" in far, not the "e" in men.

Hmmm. I think it should be a like as in cat or car, but when speaking fast it's easier to say "uh". The proof is that when speaking slow, nobody says "uh", only "ah". Hope this helps.

Of course, there are others who pronounce it "ay".