Catholic Charities have some good people and some bad eggs, same as everywhere.
They are probably the best bet, because they do not exploit birth mothers. Their home studies are brutal, like all home studies are. This is just a fact of life for domestic adoption agencies, no getting around it unless you can afford $50,000 and a private broker. But foreign adoptions have a huge disruption rate, are very expensive and rife with corruption.
If they did not make an issue out of where you go to church on Sunday, taking a stand on Tradition would be very imprudent. It would be the birth parents who choose on the basis of photographs and letters describing who you are. The birth mothers love their babies very very much and want them to be raised Catholic. You would meet the birth parent(s) on a first-name basis and be willing to send along progress reports to the Catholic Charities people. In other words, being compassionate, grateful and kind to the birthmother instead of just wanting a baby. This is helpful as the kids get older and how they came into the family becomes an issue.
I am a big advocate of compassion for birthmothers.