I feel bad you didn't get more responses, so I'll post something. Yes, most adoptions cost money. So do most births, but if your birthing costs are covered by health insurance, they're largely invisible. There are really not adoption grants out there for non-special needs, private adoption.
The tax credit helps, but it's refundable, which means that 1. you don't get it until after you've spent it, and 2. you have to owe enough taxes to get that amount back. After five years, we didn't use all of ours up because we didn't make enough money.
The above paragraph should give you hope that people of modest means can afford adoption, though!
How did we do it? Savings. A home equity loan. Gifts from family members. It can seem like a big amount of money, but people regularly drop $25-30,000 on a car, for goodness sakes. (not us, but still...) Helps put it in perspective. Also, if you look at it as a percentage of the total amount you will end up spending over your kids' childhood, it's not so much either! Having a child is not a smart financial decision. That's not why we do it, right??
A high-quality agency should give you a list of all of the expenses, what they are for, and when they are due in the process. This will make it seem more manageable and give you a starting place for budgeting and planning.
Best wishes -