Can I get my civil protection order from D.C. enforced in another state?
Yes. If you have a valid Washington, D.C. civil protection order that meets federal standards, it can be enforced in another state. The Violence Against Women Act, which is a federal law, states that all valid civil protection orders granted in the United States receive "full faith and credit" in all state and tribal courts within the US, including US territories. See How do I know if my civil protection order is good under federal law? to find out if your civil protection qualifies.
Each state must enforce out-of-state civil protection orders in the same way it enforces its own orders, which means that if the abuser violates your out-of-state civil protection order, s/he will be punished according to the laws of whatever state you are in when the order is violated. This is what is meant by "full faith and credit."