Linda,
I also work a full-time job and mystery shop mostly on weekends.
With experience, I have learned which companies/schedulers will assign the shops quickly and which might leave the application hanging. That makes it easier to rely on the former. If they delay longer than expected, I quickly follow up with an email.
If I apply to one of those companies that will let the application hang, I give them a time to act. For example, "I wish to complete this shop on October 6th. However, in order to better book my schedule this offer expires on Monday, September 29th at 6:00pm."
I hope that helps.
Happily shopping Rhode Island and nearby Massachusetts and Connecticut