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I have everything all set to glue my dovetail but it is just a hair loose. I put one piece of thin masking tape on both cheeks and it fit perfect.

With it being such a small amount I had the idea of just using a piece of paper as a shim. Is this ok to do? Or will it weaken the joint?

Should be fine. I wouldn't use masking tape (and from your description I don't think that is what you have in mind), but I think a piece of paper the glue can soak into should work well.

If a dovetail is a proper fit, you hardly need glue at all, anyway. It doesn't have to do much except keep the joint from sliding apart.

Definitely not intending on using masking tape as the final solution. Was just using it to see how far off I was.

Seeing how close I was, I was about to start sharpening my hand plane to get some good paper-thin shavings. But then I wondered if I could just use paper instead.

I've found that wood shavings from planing work great. A plus is that they're already curled if you want to wrap them around the tenon.