If my husband worked out of state (South Dakota) in December, but came back every weekend, does that make us part year residents of South Dakota?

No, you (and he) are full-year residents of your home state and non-residents of South Dakota.  

You will need to file a Non-Resident South Dakota return and then a Resident return for your home state.  Be sure to do them in this order to minimize issues with calculating any "Credit for Taxes Paid to Other States" you may qualify for.

Thank you! Since South Dakota doesn't have taxes, all I need to do is put that I am a full time resident of Nebraska. Is that correct?

Correct.