Any bitcoin you hold at the time of a split, will be on both chains going forward. All addresses and coins in them, and past transactions to give those coin balances, will be on the common part of the chain. And therefore they built on separately and differently by all forks that carry on from there.
So your address pre-split exists on both chains post-split. Post-split you open the wallet in both Electrum and Electrum Cash, and as long as you are sure they are connected to their respective chains, your transactions and coin state will be for the same addresses but different chains. And they will not affect the other chain.
You do a transaction in Electrum Cash and it has the replay protection SIGHASH_FORKID in it. This sends from your address 1234567890 on the BitcoinCash chain. If it is received or replayed on the segwit-BTC chain, it will not be able to be processed (it has an unprocessible SIGHASH_FORKID which is unknown) and your balances will stay the same.
You do a transaction in Electrum and it has no replay protection in it. This sends from your address 1234567890 on the segwit-BTC chain. If it is received or replayed on the BitcoinCash chain, it will not be able to be processed (it lacks the SIGHASH_FORKID expected in all transactions) and your balances will stay the same.
Note that if there are further splits of the segwit-BTC chain, there is no replay protection there. So someone may be able to steal your coins on the chain you are not doing a transaction on.