Puddling here, too, on my 1990's Singer, and the very first quilt I ever finished was a full size courthouse steps I did all SID. Where there's a will, there's a way!
I just got my FMQ foot a few months ago and haven't been brave enough to commit with it, but I'll be doing that soon.
The feed dogs on my machine don't drop, but I had a plastic cover for them that came with the machine. I'm sure you could buy one now.