What makes my singer 758 keep getting out of time?
Speaking from experience with Singer Touch & Sew 750s, which I believe are similar:
Manually turning backwards might cause problems, particularly if resistance or "catching" is felt. I believe this model has "plastic", or synthetic gears, which were a good idea at the time since they were quieter than metal and perhaps required less maintenance. But after 50-60 years, these gears become more fragile and can (and do) break.
Replacement gears are widely available for under $20, but it can be a real "adventure" to replace and retime them. Many repair shops will advise "not worth it" given labor cost and underlying value of machine. The lower horizontal shaft and gears are visible if you remove the lower cover - if pieces of gears fall out, well, that's a good clue, isn't it! Timing can "slip" if gears are not secured well (allen head screws loose). If there is a belt drive, rather than a vertical shaft with gears, the belt might be deteriorated as well. See other related questions and answers on timing 750 T&S class and availability of service manuals.