Just to add to dodobookends answer, yes, you really should search all the possible people that it could be. One reason for this is that if you can eliminate one of them then the other is more likely to be the person that you are looking for.
If you are having trouble finding birth certificates then I often find the General Register Office (GRO) website very useful.
You can search for free for birth and death certificates. You need to register for free here:
https://www.gro.gov.uk/gro/content/certificates/login.asp
and then click on 'Search the GRO Indexes' which takes you to this page:
https://www.gro.gov.uk/gro/content/certificates/indexes_search.asp
You can search for birth certificates from 1837 to 1934 (and also 1984 onwards) with a range of +/- two years.
You can use surname, first name, mother's maiden name, district of birth or any combination of these in order to find a bith certificate.
Obviously, if it is a very common name then a lot of matches will come back. This is why it helps to know the mother's maiden name (I presume you have this as you will likely be able to trace it from the baptism?).
Names can be mispelled so if you don't find the correct birth certificate then you can add the option to search for similar sounding variations (like with Ancestry and FindMyPast etc).
If you don't know the mother's name but you do know the names of any siblings then you can compare the records to find two people of the correct name that have the same mother's maiden name.
This will generally work, but there are cases when it won't. I had a situation where two brothers living in the same city married two sisters.
So the children of both marriages showed as having the same mother's maiden name. In that case, I actually had to pay to see the birth certifcates to determine which child belonged to which family (they didn't all appear together on later censuses).
In that case you can get a digital image of a birth certificate for £3.50 direct from the GRO website.