I am representing myself in a divorce case. My husband wants to sign the papers and settle out of court. Can this be done?

Absolutely. It's very common in cases for the parties to decide they want to settle. You and your husband sit down and work out what the final order should say. Then contact the clerk of the court where your case is filed, ask them how to get in front of a judge to enter agreed final orders. There may be a waiting period from the date of filing (here in WA it's 90 days), then once that's out of the way I'm sure any judge would be happy to have a case be resolved peacefully.

Good luck to you both.