can I send in my current Mac after I receive the new one so I don't have any time without a Mac?

Yes.

Also see the following from Apple Trade In - Apple

How quickly should I return my device?

Your trade-in estimate is valid for 14 days after you receive your new device. It’s important that we receive your trade-in within this time frame to ensure that you get the estimated value.

If you arrange the trade while making the purchase on the new Mac (either online as part of the same transaction or in store) then the trade value will be credited to your payment method (e.g., credit card).

If you arrange the trade at some other time, then it is a gift card that you can use for a future purchase.