No, if someone marries your cousin, they do not become your cousin by marriage. The person who marries your cousin is known as your cousin's spouse or husband/wife, but they are not considered your cousin.
The relationship between you and your cousin's spouse remains indirect, and they are not considered a blood relative to you. Your connection to them is through your cousin's marriage, but it does not create a familial relationship between you and them.
If your cousin's marriage ends in divorce, that person will no longer be your cousin's spouse, and they will no longer have any legal or familial ties to your family. As a result, they would no longer have any relationship with you, either, as they were never your cousin or relative to begin with.
In summary, marrying your cousin does not make the person your cousin, and any familial connection with your cousin's spouse is entirely dependent on your cousin's marriage. If that marriage ends, the person is no longer considered a part of your extended family.