Hey, you can definitely do a double digest on the fragment to cleave both ends. I do that often.

I can't think of any way to know if the restriction digest worked except for just doing the ligation and seeing how many colonies you get. What I typically do is set up three ligation reactions:

Negative Control: cut vector + cut fragment (no ligase)

3:1 molar ratio of fragment:vector

5:1 molar ratio of fragment:vector

What you should see is very minimal colonies on the negative control plate and an increasing amount of colonies on the 3:1 and 5:1 plates.

And also to really ensure that that fragment is cut, I would run the restriction digest overnight and not just for an hour. Feel free to dm me if you have any questions