If I write:

Don't!

And up arrow onto that line, NVDA will read simply the word "don't", without articulating either the apostrophe or exclamation mark.

That happens with punctuation level set to none, some or most.  If I have punctuation level set to all then NVDA reads "don tick t bang".

If I right arrow through the word, then NVDA will always read "d o n tick t bang" regardless of the punctuation level

Given you worded them "apostrphe" and "exclamation" have you edited the "punctuation / symbol pronunciation" or dictionary?

In the punctuation / symbol pronunciation dialog, there are two entries for each.  Press the exclamation ! to jump to each of its entries.  Press apostrophe ' to jump to it's entry used when it is on its own.  The in-word apostrophe is three above the other one.  All four are set to symbol level "all" by default.