What would you recommend I use instead as a reliable and hassle free alternative for the next couple of weeks until my cold wallet gets here? I'm looking for something that's both reliable and won't take too long to set up/require identification/etc.
Non-custodial wallets
should not require identity verification. That is for centralized exchanges and custodial platforms, like the one you are using currently, Binance.
I'm actually invested way more in ETH than I am in BTC.
It is going to be hard to give you an advice without knowing how much in question you are trying to move. I would hardly advise anyone to move a huge amount of money into a hot wallet
, neither hodl it in Binance (that's even worse). So if the assets do not cost so much, you can use Electrum for your bitcoin and Unstoppable wallet for your altcoins.