Have the line snaked to make sure it doesn't have a partial blockage of lint or something that's slowly building. Maybe sewage sludge is partially blocking where this branch joins the main?
Either way, at 60+ years old for likely what is galvanized drain lines, your options are slim, and you may be reaching the end of the life for that line. Yes, that sucks, and will take a lot of dirty work and time or $ to replace, but it eventually will likely be required.
Why can't you drain the washer into the utility sink? Why not into a larger one?