Hello. I've seen people asking managers in signature campaigns to change the payment addresses and explaining that the reason is very high transaction fees. That's why I got a question: if one changes the address, does it really make the fees lower? The current fees are not that crazy, but $10+ still seems high to me. If I change the address, how much the things would change and why? I thought everyone is offered approximately the same fees. Moreover, during a short period of time one can still be offered slightly different sums regardless to the wallet address onto which the money came.
Thanks in advance.
Not even the slightest. Changing the address has no effect on the protocol as on the protocol level, addresses don't even exist, it is all about UTXOs.
So no, using the same address or changing it will have no effect as the address is simply something for humans to understand, but for the blockchain it only matters to reference transactions where you got coins.
It does make a difference if you chance your adress to a segwit compatible adress. They either start with bc1... or 3..... ( 3 could also mean multi-signature so be aware.)
https://bitcoin.stackexchange.com/questions/62437/requirements-to-accept-and-send-segwit-transactionsWhy they are much cheaper then normal transaction is explained extremely well over here,
https://segwit.org/what-is-behind-the-segwit-discount-8515a8d3bca9Note that you probably still need to make a network transaction to send your bitcoins to a compatible adress before you will see lower fees.