The crystal can still be the cause of your problems, which might be hidden now by a better DS1307 which is less sensitive to the poor crystal characteristics!

My recommendation:

1. Use the best crystal you can get, not necessarily the cheap one from your local shop, and

2. Add the RC-filter Michael suggested. Remember, that the DS1307 has a buit-in switching mechanism, which switches the supply from Vcc to Vbat and which can be interfered by a too fast fall time of Vcc, resulting in a momentary undervoltage condition (voltage dip) and a stopped oscillator.

Kai