Here and there you can find transcriptions that use what you might write as [ɒ] or [ɔ] for these words. For example, Merriam-Webster does so as you can see and hear in their palm and calm. There are two possible vowels there, one rounded and the other not so, and the coda’s "dark" ‹ɫ› may or may not be present.
But at M-W they use a weird and nonstandard transcription system, for example giving palm as [ˈpäm], [ˈpälm], [ˈpȯm], [ˈpȯlm]. That means that they write ‹ä› for unrounded IPA /ɑ/ as in their father, and they write ‹ȯ› for rounded IPA /ɔ/ as in both their cloth and their thought.
(Do remember though that there is no meaningful distinction to be made between [ɒ] and [ɔ] in American English; they’re just random allophones of phonemic /ɔ/.)
I think you and I share mostly the same accent, but I’ve never heard rounding on start /stɑɹt/, [sdɑ˞t] that I can recall. Not the way there often is in Chicago [ʃ(ə)ˈkʰɔgo(w)] when it has the same stressed ɔ vowel as Milwaukee [ˈmwɔkɪj] has. There are some older dialects that round Ma and Pa and Grandpa, hence the Maw and Paw and Grampaw eye-dialect spellings of those you sometimes see. But I don't remember that happening in the Chicago area.
"Dark L" in the coda is well-known to cause rounding, as in fall, mall, all, call, tall, Paul. The rime for all of those should be the same [ɔɫ] for us. So our palm vowel is rounded like in our Paul, just as our calm vowel is rounded like in our call.
Both just add the /m/ at the end of the other version. So call is [kʰɔɫ] and calm just adds [m] to make [kʰɔɫm], but the [ɫ] may or may not always be there. The very same thing happens with Paul and palm.
All this is completely normal for the Chicago area in particular and perhaps running north of it for the Upper Midwest in general. That’s what I think that Merriam-Webster is saying there is normal for American English.