Yes, with Primal you have to give some details. Any wallet that you can buy sats with will require kyc.

You could use a different wallet, such as Wallet of Satoshi that only requires an email (create one unconnected to you) to login if you sign out. It is largely anonymous as you get a random lightning address. You could send the sats from Primal to Wallet of Satoshi cheaply, giving you some anonymity.

There are probably many more ways to do this better, I am in no way an expert, I've only learned this myself over the last few years. There are many incredible people and resources on Nostr, just keep asking.

Oh, and use a VPN too, that'll give you some further anonymity.