Try warming the prey in hot water. I use the hottest water from the tap I can get. Test the prey by slightly squeezing the skull and stomach. You should be able to feel if the inside is hot, cool, or cold. You want to present the prey hot to represent a live mouse. Since a mouse has a body temp around 100F the f/t prey needs to be very throughly warmed to give the correct heat signature.
I place mine in zip bags but they still will occasionally get wet. I will dry them a bit before presenting them but its not an issue. Especially if you have a paper type substrate.