Does the FAA maintain pilot records from before 1998?
If you took a checkride the FAA has your 8710 (form you would have filled out) which will have your flight times and can be used as proof of time.  Follow these steps to get your records.

You can search the airman database.  My father's and grandfather's records are in there from the 1970s and 1980s.  This just lists certificates though, no times.

You can also request a replacement certificate.  If you are a PPL then they will send you one.  You'll need to do this anyway to get the new plastic card.  Also, no times...just a certificate...which prove that you have at least the minimum flight time required to hold the certificate.

Good luck.