Much depends on your tastes in 1) lodging and 2) food.
We can do a 100% camping, 2-3 restaurant meals per week trip for four on $80-90 per day. We did eat a fair bit of "fresh yesterday" bread, and paid $20-$30 per night to camp in NY state parks (with showers).
You may find food a little pricier, but you should be able to camp more cheaply. But sharing a site three ways makes that a relatively small part of the budget. Your big item is food.
You need 4000-6000 calories per day (depending on your size, weight, load and daily mileage). You aren't cooking out of your own kitchen. You might be able to get away with $15 per person-day, but it won't include any restaurant meals, and you're talking little or no prepared food. Think about what you normally spend on food. Now double that for the extra you need to eat. And adjust for any differences in how much is restaurant vs self-prepared.