My K-1 came with K-1's for 4 different states - I purchased the states, but it won't allow me to enter the K-1's on the state return. Has anyone had to do this?

Start a return for each of the states, either resident, part-year resident, or nonresident, as appropriate.  For the nonresident states, there will be a section for you to designate what portion of your total federal income is related to that state.  You'll use the amounts from those state K-1s to populate the income or loss associated with that particular state. 

To the extent you pay taxes in those nonresident states for K-1 income sourced in those states, you'll get a credit on your resident state tax return for "taxes paid to another state".  That way, you don't pay tax more than once on the same dollar of income.  TurboTax will do this automatically if you prepare the nonresident states before you complete the resident state tax return.