Cash is legal tender. Plastic is not. If I offer you cash (in reasonable denominations) to pay a debt and you refuse it, then the debt is cancelled. That's the law in Canada. Canada inherits that from British common law -- just like the US but with a later fork. This is an ancient principle, so it's likely to be the same in the US unless there was an explicit change.
In this case (hair done), the debt is incurred so you don't even have the excuse of forward negotiation.
For completeness: The debt can be incurred before the product is taken. If you run the items through the cash register and say "Total: $19.56", I now have a functional debt. If I hand you a $20 and walk away, the debt is paid.
Of course: IANAL .. YMMV.