The adoption industry has had many motives to change not only one's name, but DOB (up to as much or slightly more than a year), and even place of birth, to even a different state!!! This makes a search nearly impossible.
My own informal survey seems to say that DOB and place of birth are changed in 25-50% of the cases. but this high a number may be limited to selected localities. So if you if you want to find someone, don't depend entirely vital data being exactly right.
Some adoptees find it hard to believe their DOB is not what they have celebrated all their life. But this emotional difficulty must be overcome, if one is to find a birth parent.
Now, the courts can order dates and places and names changed, but they cannot change your DNA. So the way to find someone is to match blood DNA. However, this costs money. So before you spend it, you need real convincing proof you have the right person, before taking such a test.
Now if two persons are searching and should be looking in the same registry or bulletin board, they can find each other if they follow the following procedure. This procedure is best described as building on your successes.
First, you can use an approximate date to find someone, provided they are also looking. Unless you are in a state that has open records you can only use internet bulletin boards and registries or pay outrageous fees for someone to search. But searchers are "lazy" and usually only look for an exact DOB match. They do not "dig" to find out if the DOB and place of birth was changed. (Such digging costs real money, not the mere $20-100 they ask for, and which you are likely throwing away, and you can look in the same databases as well, usually for free.
Second, if you find someone who has a birth date within about one year of when you think the birth occurred, you can explain your situation to whom ever you contact and start asking about inheritable traits. (Tell this person that DOB and place of birth do not have to match closer than about one year since these are very often changed anyway.) Also, remember, only about 3% of the population is adopted, or 1.5% of each sex, and so if you get with in one year, you can know that this improves your chances significantly. But this could still leave tens of thousands of persons.
Third, if you come up with 20-30 inheritable traits, then you have at least a "lead". Now questions for inheritable traits include hobbies, careers, natural skills, things people like to collect, names of pets, family members names and the list goes on and on. This would justify exchanging photos and full-page handwriting samples. Now you could be down to just a few thousand possibilities, if you see to find enough in common.
Fourth, photos and handwriting should be compared between all blood relatives. There is a handwriting analysis book for $7.00. The analysis is very easy. If you come up with about a 90% match you have match. (You are not trying to determine what personality two persons have, only that they are nearly identical.) As for photos, you look for eye and hair color, ears and nose and mouth shapes, jaw and chin bones, etc, etc. Keep in mind that children have mixtures of features of both parents. If you have some good photo matches between various blood relatives and some handwriting matches, you could now be down to just a few hundred possibilities.
Fifth, consider any inheritable medical problems, including vision, hearing, heart, spinal, mental, disease, etc. Also consider blood types.
Sixth, consider any circumstances surrounding the adoption. Are there any similarities. If any circumstances are contrary, but are the kind that agencies, courts, lawyers, etc, could change, then ignore them. Only pay attention to what matches. (What does not match can be resolved later.) By now you could be down to just a few possibilities, but you could also have overwhelming evidence by now.
Seventh, if you think you have a "match", then you can always do a maternity or paternity DNA blood test. This can even work to some extent for siblings. But before you spend the money, be sure to have done as much investigation of matching traits and so forth so that you have every confidence the DNA test will prove to be a match. Try to leave "no stone unturned" in all your investigation.