Section 5(v) of the Hindu Marriage Act, 1955 prohibits marriage between individuals within a prohibited degree of relationship, which includes first cousins. This law is in place to prevent consanguineous marriages within specific degrees of relationships, citing genetic risks among other reasons.
Moving to another place within India or another country might not solve the legal issue, as most jurisdictions uphold similar regulations on cousin marriages due to similar concerns. However, some countries and states permit such marriages, like certain states in the United States, parts of Europe, or other regions globally.
Regarding the same "gotra" issue, it's traditionally discouraged in Hinduism due to belief in common ancestry, though its legal implications vary.
For resolution, seeking legal counsel to explore the possibility of a Special Marriage Act, 1954 or adopting alternative legal mechanisms, if applicable, might be an option. Nonetheless, navigating family sentiments, societal perceptions, and legal constraints will be complex. Overcoming familial objections might involve counselling or legal mediation.