Do they want full payment or just a deposit for now? If there is language in the contract that suggests you've already paid the deposit, I could understand them wanting the deposit before they sign. If that's the case, could the wording be changed so that they feel comfortable signing first and then receiving the deposit (though still puts you on the hook for paying the deposit)?
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Especially for full payment, I would request that they sign first. With both signatures, you are on the hook for the money and they are on the hook for the service. I agree that I would be hesitant to hand over the full amount before both parties have signed the contract. Listen to your gut feeling on this. If they're not willing to sign without the full amount, and you're uncomfortable with giving the money first, then I would keep looking for another photographer.
When is your wedding? Most photographers don’t ask for full payment until right before the wedding, I wouldn’t be comfortable paying the full amount a year out (or however long). What if something happens or you want to cancel with them, do you get the money back?