Signing a Unconditional Waiver when we have not been paid

First, note that Virginia does not require preliminary notices on public projects. Meaning, a prelim wasn't required in order to preserve the right to make a payment bond claim. 

As for the waiver timing - that's tough. Virginia does not prohibit the use of unconditional waivers before payment is made, so giving an unconditional waiver before being paid will relinquish payment bond claim rights. I suspect you already know this, but there's really no reason to require an unconditional waiver prior to payment - a conditional waiver is just as effective as an unconditional one as long as payment is actually made. And, if the GC also understands that conditional waivers are perfectly effective, then their refusal to use them may well be a red flag. This article should be a good primer on the issue: Do I Have to Sign a Lien Waiver to Get Paid? 

One option may be to draw up an unsigned unconditional lien waiver and show that to the GC to try and induce payment. If they know you'll sign the unconditional waiver right then and there when they hand over your check, they might be more willing to work with you. Another option would be to threaten to make a bond claim - more on that here: Notice of Intent Before Making a Construction Bond Claim. Or, if the customer is still resistent, actually following through with a VA payment bond claim or legal claims (like breach of contract) could be an option, too. 

Finally, note that submitting the waiver is always an option, too. If you can get the contractor to agree, in writing, to provide payment immediately upon receipt of the waiver, that could be reassuring - and it's a reasonable ask if an unconditional waiver is being demanded. If the customer refuses, that could be yet another red flag and provide a sign that a payment bond claim may be necessary.