All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, we supplicate Allaah to reward you with the best for embracing Islam, help you and us to remain steadfast on the truth, and increase your aspiration to do good and learn the rulings of the Sharee'ah (Islamic jurisprudence).
The matter is as you were told that adoption is prohibited in Islam since it leads to the mixing of lineages and losing the rights, like inheritance and the like. Allaah Says (what means): {And He has not made your claimed [i.e. adopted] sons your [true] sons. That is [merely] your saying by your mouths, but Allaah says the truth, and He guides to the [right] way. Call them by [the names of] their fathers; it is more just in the sight of Allaah} [Quran 33:4-5]
Even if an adoption has happened, it is compulsory to re-establish the true lineage if it is possible. However, if you fear harm by being traced back to your true lineage, then you may adopt a common ascription such as, the son of 'Abdullaah or the son of 'Abdur-Rahmaan which does not indicate tracing back to a specific family. Salmaan Al Faarisi who was a companion of the Prophet used to say: "Islam is my father, I do not have any father except Islam." Thereupon he was called 'The son of Islam'.
It should be said that the religious rights of the parents such as being kind and dutiful to them, and taking care of them, do not cease, in any case, even if they are non-Muslims.
Finally, we advise you to befriend pious people and attend religious gatherings held in Islamic Centers. We also advise you to benefit from useful Islamic sites.
Allaah Knows best.