Is there a better way to move things out of trust wallet (ethereum based tokens and bnb ones too) without paying crazy network fees to move them (in particular Ethereum)
You have to go through with the exact process that you're thinking of avoiding. That's how the network goes, you have to pay for the existing fees.
And if the network says that you have to pay a quite high fee for your transaction to proceed and that's the suggested amount, you have no means of avoiding it.
But there's a way if you really want it, wait until the network settles and becomes quite empty or shows a low suggested fee before you proceed.