How to keep data anonymous across two linked surveys

Hi there,
to ensure response data is stored separately from contact details, I have linked up two surveys. The first one collects response data, the second contact information.
Unfortunately, there is still a possibility to link contact information to response data via the (start date and end date) timestamps that are stored for each record. Is there a way to turn off the recording of these timestamps for one of the two surveys, e.g. the second one? Or are there other suggestions how to decouple both surveys more effectively?
Any help is much appreciated
Ingo