Contract Question

I might be overthinking everything but do you all sign a contract before you send it to a client when you get a signed contract back then sign it and send them a copy?

I over thing everything but my thought process is say I signed contract "A" and sent it to client "A". Then I never hear from them and then a nother client "B" wants the date and they pay and send you a signed contract "B". About that time the first contract "A" comes back with payment but you just took payment and contract "B". Now you have two contracts out for the same date and time and your signature on both.

So I was thinking of sending a contract, when it is returned signed I would then sign a copy and send to the confirmed client.

Thoughts???