I know of a few services that will send money to your PayPal account in exchange for bitcoins but when I go to pay at the food site it asks me to log into my PayPal account to pay. Unfortunately it does not just provide a PayPal address.
In theory I can just exchange my bitcoins for PayPal and pay for my food that way but that seems like a roundabout way of doing things and my ultimate goal is to drop all of my bank accounts (does PayPal allow you to have an account with no bank account?).
Is there a more direct solution?
You can create a PayPal account without a bank accounts this may be the easiest option for you. Just be careful as non verified Paypal accounts have certain limits on what you can spend each month, you need to make sure that your total spending per month does not exceed these limits.
If the unverified paypal is a business account and if you slowly accept small payments of $20 for a while the account becomes more useful. I currently use an unverified paypal account for numerous purchases and probably spend around $200 every month from it.