No, you do not enter a state at the 1099-DIV screen.
Later, in the 1099-DIV interview you will encounter a place to enter the NM dividends, if any.
If your mutual fund company provided you a breakdown*, you are only interested in your home state**. Multiply the % for your state by your total tax exempt dividends to get a $ amount (you can't enter the % in TurboTax [TT]). When asked which state, check the box "I earned tax exempt dividends in more than one state". In the drop down menu, select your state and enter the $ amount you calculated. In the 2nd box, select "More than one state*" (at the bottom of the scroll down list) and enter the remaining dollar amount.
*Most mutual funds will provide a breakdown. But you usually have to ask for it, or find it on their web site.
**Your state will tax all the dividends except the dividends from municipal bonds from your state and US Territories.
For details on how to enter, see: https://ttlc.intuit.com/community/state-taxes/discussion/re-exempt-interest-dividends-are-entered-from-form-1099-div/01/1318344/highlight/false#M58913 Most particularly, see the screen shot at that link.