If the rest of the W2s are the same, complete the state parts on one W2. Use the W2 with more complete information. Each state will have a line...boxes 15, 16, 17. This should indicate the wages and state taxes collected from each state. This will flow to the correct states when you complete those returns.

If you need more help with this, ask in the comment below.

To be clear: I only have one W2. Half of it has different info than the other half (four slips per W2). Do I need a new W2 — maybe two separate W2s for each state?

No, you can use what you have. The boxes above the state part should all be correct. Just combine the information when you enter it in TurboTax. Does that make sense?

I think so, so I'm just combining the different values in the two different Box 17s? And then using the rest of the info from the more complete W2?