I guess my concern is that I will have pee that is 96 degrees, and then it drops another degree before they take it and then it's around 94 or 95 and they mark a red flag and I have to retest on the spot (which would result in failure)


Under DHHS regulations, the acceptable temperature range is 90-100 F. A sample of 94-95 F is fine, therefore the tech has no grounds to reject it.

A temperature drop is expected when a liquid is transferred (or urinated) into a container of cooler temp (in this case, the empty sample cup). Thats why a temperature range was established.

Make sure though, that the empty sample cup doesnt feel too cool to the touch when the tech hands it to you. The last thing you want is the temperature to drop so much it falls out of the 90-100 F range.
In an effort to minimize the temp drop occurring during transfer, I often "prewarm" the empty sample cup by tucking it in my armpit while I close the bathroom door, unzipping, etc.