I currently have my BTC stored on a MultiSig cold wallet, which can't be connected to a lightning node. Do you think it's a better idea to move them to a lightning wallet like on electrum for example before the transaction fees get too high? I want to avoid paying a big amount just for one transaction in the main network
If you are not spending your BTC's anytime soon, leave it in your multisig cold wallet, which is a more secure way of storing your coins. It is not a great idea to move your coins from your cold wallet to an online wallet, what is more important is that your funds are secure and probably by the time you want to spend, the mempool may be less congested and fee rate would drop.
Especially if you use Electrum wallet you will save some gas fee. Because the gas fee is much less in Electrum wallet than other wallets.
There is no gas fee in BTC, it is called transaction fee in the BTC network. Take note that tx fee is paid to BTC miners and the fee you pay isn't determined by the wallet you use, instead your tx fee is determined by your input and output and how congested the network is.