For instance 早々(As Soon As) has two readings: そうそう, はやばや. Does it mean both are equally acceptable? Something to do with dialects? One is used when it’s a suffix the other when it’s a noun?
This somewhat depends on the exact vocab word. This is one of the persistent issues that’s been around since the beginning. Usually two different readings will have two different meanings or nuances. There are occasional cases where both are fundamentally the same but usually one reading is preferred.
In this case, a thorough answer can be found here: https://japanese.stackexchange.com/questions/51902/how-should-i-read-早々-as-そうそう-or-はやばや
My experience has been that more advanced words that have different readings also have quite different meanings. I can’t think of any at the moment, but in my N3/N2 class, we’ve seen about 10 or so. Basic words however tend to have the same meanings but with older pronunciatuons. I’ve asked some ppl here in Japan about these simple words and get the response that one pronunciation is definitely used more than the other. While the other one is correct, they often have to think about it for a while or look it up.
Jisho.org will have separate posts for words that have different pronunciations and their meanings. So far, it’s been 99% correct. I can also recommend tangorin.com as well for the same thing, but they have more example sentences.