...why not just use a decentralized P2P exchange like LocalCryptos instead and get your funds directly deposited to your cold storage?
To be honest, since I'm still new to it I was was a bit nervous about using a p2p system like that. LocalBitcoins has been suggested to me, but I understand that is still centralised and you still have to pay a fee to transfer from that. What security does LocalCryptos offer a buyer? Reading their site it simply said "Once the seller confirms payment, crypto is released from escrow to the buyer" but doesn't that rely on the seller being genuine?
The whole process is pretty secure. The escrow isn't a middleman or something like that, just thin of it as a temporary "bank account" that hold the sellers assets. If you see an offer that suits your price point you just send an offer to the seller which in turns evaluates it. If indeed it accepts the offer he gets the money and you get the equivalent crypto in your wallet.
Remember that, as a buyer, you should never send money to the seller before he/she deposits money on the escrow account. If the seller tries to run away with your money (by denying that you ever sent it in the first place) you can just raise a dispute with him. An external agent will then check the deal ( verify exchanges, see the conversation , etc) and the crypto/money is returned to whoever is right.
As always, try to make a deal with a reputable seller with a > 99 % positive feedback