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 mean by move is to transfer right? That's impossible without paying fees. Every network has it own fee, some are cheap and some are bit expensive but you can't move any tokens without paying it. All tokens/coins have that standard when it comes to fee. Nano/xrb before have almost zero fee and damn fast. But I'm not sure if this is the case until now.
But there are still other options such as matic and fantom which clearly have low fees.