Hi Kay!

I am including a excerpt (and multiple links below) with more information on transplanting cactus and specifically Euphorbia. Yes, the short answer is that by allowing the cutting to dry for a few days, you can help to prevent a common issue of fungus infecting the cutting. One thing I would be mindful of is the temperature during your move. While it may not hurt for the cutting to dry out a little, leave the cactus baking in a hot car after a day of travel could be detrimental.

Quote: 

The primary problem with cactus cuttings is the development of fungal soft rot. This condition begins at the cut surface and eventually reduces the entire cutting to a slimy mass. Two techniques are recommended for avoiding this problem. The first is to simply allow all fresh cuttings to sit in a warm dry place from one to 14 days before they are placed in the rooting medium. The larger the cut surface, the longer they should be allowed to dry. During this time, the cut, moist surface will form a dry callus that is far less prone to rot. Although leaving a new cutting exposed for several days may seem like a radical idea, remember that a cactus is able to survive periods of drought. As long as the cutting is not noticeably shriveled, it is probably okay to leave it dry. Newly cut stem segments may also be dipped in garden sulphur before planting to prevent the onset of soft rot. Do not water the cutting for a week, then water sparingly.
Source - https://aggie-horticulture.tamu.edu/databases/cactus/growingcactus/

Additional information:
https://www.ndsu.edu/pubweb/chiwonlee/plsc368/student/papers01/pneisen/Propagationofcactiandsucculents/cactuspropagation.htm

https://www.canr.msu.edu/news/secrets_to_success_when_propagating_succulent_plants