With electrum you keep the same addresses. Once you start the software, it will ask you if you want a legacy wallet or a segwit address, and then you get a wallet file, and this file will give you access to your coins, much like Bitcoin Core's wallet.dat, and also you will get a seed which can generate the content of this file anywhere you are at without holding the file (so save it somewhere that nobody can see it or better memorize them and destroy it).
Will electrum allow me to send coins to multiple addresses in a single transaction with a SegWit address, just like I do with my Legacy address?
In bitcoin it's best to be paranoid about security so don't use that address anymore and consider it compromised.
If it's really compromised then whoever holds the private key can sign messages also.
I won't be using that address anymore, but to make everyone aware of the same, what should be done? Making a thread in Meta and signing a message with that address saying that its private key was used on a website and I believe it's now compromised, so don't send any coins there if someone asks anything by signing a message from that address. Will it do?