I've been using
this manual. I'm aware that there were a bunch of different versions of the Model A, but this one's pictures look similar enough and nothing seems to be missing. According to the serial number and
this website, I think it was produced in 1919, so nearer to the end of Davis manufacturing for Sears and Roebuck. As for needles, a modern needle seems to fit in perfectly, although I would love to be corrected on this. As soon as I got the machine I oiled every spot as indicated in the manual and let it sit for a day. Putting upward pressure on the take-up lever doesn't seem to help. I'm still working on the pictures, unfortunately, it's mostly an issue on my end and something or another about security tokens.
Either way, I wanted to get this reply in in case it helps and say thanks for your response!