Take a look até
http://mempool.space/ to get the fee rate you need to use, this is the best site available for that.
Now you need 22 sat/vbyte for a confirmation, which is not huge.
Not sure how reliable is this one:
https://bitcoinfees.earn.com/But it's currently showing the fastest and cheapest tx to be at 102 sats/byte, which, for median-size tx, converts to >$5, which is quite a lot.
Rather than "fastest and cheapest", i would call it "most likely on next block". And their data seems to be out date since currently there's no TX on mempool with fee rate higher than 50 sat/vB.
I want to believe that this ordinals plague can be stopped before it grows to gargantuan size. Can NFT transactions be marked and refused by most nodes?
Theoretically possible, but i don't expect it'll happen in reality. Although there are suggestion such as transaction with big arbitrary data on Taproot's OP_PUSH become non-standard. And it's impossible to know whether people use Ordinal to make NFT or just store arbitrary data for other reason..
Or perhaps increase fee for NFT transactions?
Impossible without hard fork. Currently fee rate is only based on transaction size.