You would have to have pretty long fingers to feel anything with a finger in the doe

Just wash your hands, I do take of rings and my fake fingernails are off now

I just use oil, I keep a light oil like almond in the barn to massage udders after shaving and since everyone is getting dairy clipped it's what I have in the barn now. I insert my four fingers of my right hand and hold my arm over the doe and apply pressure with my armpit over the does rump...some may stomp their feet, but I am very quick and matter of fact with all this. So since my hand measures 6 inches from my middle finger to where I can't go in further with my thumb outside, that is where the cervix is in most of my does.
Your do is having normal progression, the thick mucous is the mucous plug, the contractions before are called braxtin hicks and is normal in goats just like with us..lots of women end up at the hospital in false labor and sent home from these contractions and lots of goat folks think their does are delivering when they are not in labor also

Also know the two problems in your does kidding are completely unrealted, so there is no reason to think her kidding will be anything more than normal.
I wouldn't be bothering her with a pelvic if you are not going somewhere. The biggest way of knowing your doe is kidding is a super tight and shiny udder (usually new folks on here distressed at how large it is and 'should I milk her out' questions) and then she kids. Try to check her, use a baby monitor, and when she actively starts pushing with every exhale, than simply do a pelvic with her laying down and if you don't have teeth and feet, than help her. If you find teeth and feet than sit back and let her push the kid out...repeat repeat repeat...
I do not use antibiotics in my herd for pelvics, our doctors don't do antibiotics for pap smears or pelvics either. And I am very careful with the use of products promoted by companies selling you stuff that we have no idea what is in them or if blood or colostrum products they sell come from tested stock. Vicki