Sole proprietor

That's ok.  It is still income.   You might want to use Quicken or QuickBooks to keep tract of your income and expenses.  You can pay bills any way you want, from cash, from checking, from credit card, etc.  

This is a question. It went in the wrong spot I think

So to be clear, as a sole proprietor,—— instead of depositing cash into the business account and then taking an owners draw, I can just keep it to use for personal expenses as long as it’s still recorded as income on my schedule C.?

Yes.   You don't take an owner's draw or expense withdrawals.  All the business income and expenses are your personal income and expenses in the first place. You just fill out a Schedule C. The net profit or loss is your income.