AMC first tries to extract the scan from the PDF file with pdfimages. If your PDF contains one image for the scan and a small image (or vector graphics) for your drawings, only the first will be used.
You can tick Edit/Preferences/Scan/Scan conversion/Force conversion to tell AMC not to use pdfimages, but a PDF→RASTER converter.
You can also consider marking your open questions from the Data capture/Manual window, selecting Scan as the background, instead of Original.

Hm. That did not work. selecting that option does not seem to change the outcome. By the way, when I change a preference, it seems that I have to restart AMC before the preference is honored...

This is what worked. I used convert (from imagemagik) to convert the pdf to jpg, and then converted the jpg back to pdf. Then, when I run AMC on the file (with or without the box being checked), then it works - the checkmarks remain in place, and are graded correctly.

So, for now I have a workaround, but I would really like to know why this is happening, so I can find a better solution ...

I do not want to use manual data capture, because I plan to use this for commonly graded exams with many graders and 500-1000 students ...

Thanks,
S.M.

and then converted the jpg back to pdf

This should not be necessary: AMC can handle JPG files.

selecting that option does not seem to change the outcome

Yes, this is a bug. I corrected this in hg:b3e7eed19206. This is just a line to add to AMC-getimages.pl, you can add it yourself to test.

Thanks for your help. I am about ready for my first trial of this system. I am hoping that it will help to speed up the hand grading of many exams.