Your question, and feelings, are shared by many as there are tens, if not hundreds, of thousands of Mexicans and South Americans who have ancestry similar to yours, many of whom live in Texas and the Southern United States and California today. Almost all share the commonality of varied Jewish customs, often performed in a secretive manner. These customs range from lighting Shabbat candles to a major cleaning in the spring before Passover, to salting meat before its consumption. Many only allow marriage within a certain circle of people.
Most of these people have no idea why they observe these customs, and do so because it’s the family custom for generations. A minority, however, actually have a family tradition like you have. They have been told they are Jewish and have been so since the Spanish Inquisition.
In cases such as yours there is a definitive consensus among the contemporary decisors of Jewish law. They have concluded that you unquestionably have Jewish roots. You should, indeed, be very proud of your Jewish past and ancestors and what they went through to preserve their Judaism. They did so despite the danger and antagonistic surroundings they were found in.
This glorious past, however, doesn’t paint a clear picture of an unbroken chain of Jewish lineage back to Spain. Halachically, you would need to trace a matriarchal chain of Jewish daughters back for some 600 years. Living in Christian communities for so many centuries introduces a modicum of doubt into the reliability of that chain.
We therefore conclude, to erase any room for doubt, the conversos need to go through an act of conversion with the full acceptance of all the mitzvos in the presence of three rabbis.
Although a usual request to convert is met with being put off a few times to ensure their sincerity, a case like yours is different. You would be accepted with open arms.
This conversion would connect and solidify your glorious Jewish past with a bright Jewish future. You will make your ancestors very proud in your search to reconnect to what they up so much for!
Rabbi Yerachmiel Fried