You need to set in place some guidelines because this will happen frequently and if you have your own rules it will make it easier. In my email back to the prospective renter I tell them this:
The reservation is finalized only when payment is received, so prompt payment will best assure that your lodging arrangements will be secure. We have put into place the “first pay, first book” policy in order to avoid the drama of competing claims for a given time on a property. It does not matter who might have said what, when, or what might have been understood. The first paying party receives the booking.
When they confirm that they in fact do want to book the property, I send them an electronic contract and say this:
Please find this lease/reservation I have prepared for you based on the information you gave me. Please check that it is correct and sign/initial within 48 hours. Please note that we continue to market the property until the deposit and signed contract is received. Reservations are not booked or guaranteed until the lease and deposit is received.
So, they have 48 hours from the timestamp on the email to send it back with payment and signature. If I have another that is interested, I tell them of the 48 hours the person has to return it and if they don't the other person automatically gets it. If I have someone that expresses interest and the contract is still unsigned, I will normally send them a reminder email letting them know someone else is waiting and they normally sign it asap.
I think the bottom line is: You need to set a time limit and not let contracts be floating out in space waiting for people to return them. You can also require credit card or paypal payments for the initial payment to lock in the booking instead of waiting for a check. Same for the contract. Email it and require it to be scanned or faxed back or you can use an online signature contract which is immediate.