Oh @Temple, you poor love. If it was me I'd go straight to my GP and tell them that that your emotional response to cancer has finally emerged in the form of this phobia. That you want to complete your treatment but that you were going to need some help in the form of some anti-anxiety meds in order to do so. It will only be for a few weeks, and I can't see why you wouldn't be given a prescription.
You will of course get used to it a bit once you're underway. Humans do adapt to all kinds of weird things. But honestly, don't beat yourself up about reacting like this. Cancer bloody sucks on many levels. Get some pills, get through rads and then put it behind you.
The biggest of hugs, K xox