Does This 1937 $2 Qualify For Unc?

It is really difficult to assess your notes from the images you provided. I would be very tempted to smell your note (to try to detect bleach odours). The reason I write this is due to the light look of your note plus missing planchettes (which will be visible on an original UNC note). That's my problem with the images you're posting (note just looks light/missing planchettes & looking processed). It may have been AU & the previous owner placed it under an iron? Something has removed/or severely lightened the dark green planchettes that are now faded on your note.

On an original UNC note there should be a numeric impression on the back of the note where the serial numbers are. This is called embossing & I would be very surprised if your $2.00 note has this feature (the first thing that vanishes after being pressed). You have to darken your room & put a flashlight up to the side of the note to detect embossing.

Here's what a 1937 $2.00 SPECIMEN looks like:
https://www.coinsandcanada.com/bank...m=3#banknote

So I would say it looks AU to apparent UNC.

EDIT:
Here's an original AU $2.00 (note the green planchettes around left serial & on bottom border) I posted a few years back. This note graded AU 55 (by BCS) due to some minor handling on the top border.

Try lighting the bill like this:



*borrowed from SteveInTampa over in the US note section.

The lighting trick is cool!