My next door neighbor always did those things for no-kill animal shelters. We just wrote checks for him to give his preferred shelters. Also wanted him to look good bringing in donations. My other charity was the nuns at our high school’s pension fund. When the last of our teachers died, I stopped those contributions so the pups did better.
With a lot of animal shelters these days you can “sponsor an animal” which means when you donate you say say you want to sponsor a pet and that goes toward the pets adoption fee.
Some pets have a $0.00 adoption fee when people go to adopt them. People do this a lot with older animals so they get adopted faster.
Ask your local shelter about doing that.