There are things I wish to buy on this site, yet when I try to make the purchase, I'm only given the option to use PayPal.

However, I do NOT trust PayPal to keep my information safe. Is there a way to make it so I can make purchases without PayPal?

And when I try to "pay with card through PayPal", it still tries to sign me up for an account.

If you're not seeing a separate "pay with card" option, the creators of those particular projects probably take payments directly (instead of going through itch.io). You need to ask them.