I'm sorry that you're having problems with your Climara patch. Some women are sensitive to them and it's possible that estrogen patches aren't going to be a good match for you. You could ask to try Vivelle Dot as some women can tolerate that brand and not Climara. Instead of being changed weekly, Vivelle Dot is changed twice a week.
If the patch you had on wasn't properly adhered to your skin, it could be that some water seeped under it and caused the irritation. I've had that happen with my Vivelle Dot. Here's a link to an article on how to be sure the patch is properly stuck on (and stays put!): How Can I Keep My Estrogen Patch On Until It's Time To Change It?
Would it be possible for you to pick up a couple of samples of Vivelle Dot from the doctor's office? I'm confused as to why the on call doctor can't write you a prescription for an estrogen pill rather than a patch.
You can, of course, try going without the patch until you can see your doctor. Early in recovery not everyone experiences bad menopause symptoms. It will depend on how quickly your body depletes the hormones made by your ovaries.
Some members have posted that they have gotten relief from patch irritation by rubbing on some over-the-counter hydrocortisone cream after the patch has been removed. You'd have to relocate the next patch to avoid the area since it's not going to stick in an area that has lotion, cream, oil or moisture of any type.
Hang in there and good luck at your appointment. Let us know what the doctor says, okay?