The best way I can think of to do this is to use a Wacom-type tablet to write the signature straight into Adobe Illustrator, then insert it into your drawing as a *.ai file.
DISCLAIMER: I don't use Illustrator, so haven't tried it!
Ian's method would be much better than a scanned signature. If you can get everything including logos to be vector format, then your files will be better quality and smaller in size.
However, those digitizers are awkward unless you get one where you can actually see the signature on the pad while you sign. It is difficult to look at the computer screen while you sign a tablet.
As a professional engineer I have to seal and sign drawings. I'll share what I came up with after a lot of trial and error, to hopefully save you some trouble:
I found an app on my phone that allows me to sign on my phone with a stylus (Handwrite Pro). I paid the $5 for the save to pdf option. Then I transferred my signature to the computer.
Now my workflow is to print to pdf, then in the pdf application (Bluebeam Revu, but Acrobat Writer will work also) I "stamp" my signature onto the pdf. The signature is vector and is thus very clear.
You can see an example of a vector signature on Wikipedia:
Welcome to the forum. I've done something similar, but the ones I use are either GIF's or JPEG's. I'd suggest either trying a different file type, or the PNG's might work okay with different settings when you scan the signatures.