Is there anything I can do here?

Make a spreadsheet and work it out yourself. Did you or did you not underpay tax? Can you get to their calculations?

Ignore NI. Just look at your P60 and P45 if you changed jobs. Are they right or wrong?

I just hate the fact they're saying I underpaid after having contacted them multiple times before to ensure I was on the correct tax code.

Don't get emotional about it. When they say you "underpaid" tax they aren't accusing you of doing anything wrong, and they aren't going to penalise you.

PAYE is designed to get taxes mostly right, most of the time. And for most people it does and unlike the majority of the world, we don't have to complete tax returns as a result.

Under or overpaying tax is incredibly common and HMRC are really reasonable about it.

I'm an accountant and I usually overpay or underpay by a grand or so each year. It's no big deal.

I understand it can be frustrating if you're trying to get it right first time and struggling to do so, but like I said before - lose the emotion. When HMRC says you underpaid tax they aren't accusing you of doing anything wrong. It's just a normal part of the expected process for them, like a waiter bringing your bill after you are a meal.