Marlin.45 - a sheep eats about five small square bales of hay per winter, so she won't be much of a drain on your stores. Can you use binoculars to see if she's tagged? I think you've done all you can to find her owner, so now she's yours. As she has crayon marks on her back she's probably in lamb, so needs good forage to survive and produce healthy lambs. My feeling, but others will probably disagree, is that a ewe, a flock animal, who is prepared to go off on her own to protect her unborn lambs and find the best food supply she can, is a sheep to keep.
If you really really don't want to keep her, then find a new owner. If she's not tagged then you need to get official advice as to what to do. If you don't have a flock of your own but want to keep her, then you can easily get a flock number and order a set of red eartags for her.
Sounds like she needs a health check so you need to work out how to catch her, remembering she's in lamb, so she needs to be stress free.
I don't think you would really leave the gate open and let her wander off....
ps - post and rail is not adequate to keep sheep in, unless it's backed by sheep mesh.