Given your concern towards privacy and anonymity in your post, buying your coins on a CEX and directly moving them over to your wallets would be an easy catch to track so instead of that, you can consider making use of a coin join to mix your coins and then move them to a cold storage for your safety.

That aside you can make use of a good DEX and I'll recommend exch.cx based on their reputation on this forum and I have also used them a few times too.
Also depending on how much your coins are, I suggest you use a different device to create a lightning channel and not the device you used for your cold storage.