Hi everybody!
Thanks so much for responding with advice. Here is an update...I took my Bernina to the dealership where I purchased it and had the service done. Instead of the technician working on it, the sales lady did. The best thing is, drumroll please, she got it working again! I'm so pleased because I loved my Bernina when it worked and it is a workhorse. I'm not sure what exactly was wrong because she worked on several different things and voila, beautiful stitches and no bird's nests.
For the Singer, I'm thinking of trying the bobbin washers and see if that works. I also bought all new bobbins for the Singer to see if that works. Ahhh, but at least I have my Bernina back.