Josh, welcome to the site. In answer to your question, a distinction needs to be made between normative Halachic practice (aka Jewish law) and streams of Judaism that do not consider Halachah as binding (like Reform Judaism).
I am not an expert in Reform conversion or synagogue standards, but my understanding is that the Reform community will welcome you as a member in a synagogue and allow you to participate fully, since they recognize both their own conversion (which you probably underwent as a young child) and also recognize as Jewish, or are at least open and welcoming to, those who are fully involved in Jewish life (in Reform Judaism, someone who has had a Bar Mitzvah is considered to be a prime example of this done right).
In order for you, as an adopted child, to be Jewish according to Jewish law, you would have to either be descended biologically from a line of Jewish women (yes, even if you never knew your biological mother), or convert according to Jewish law. Whether and how to convert is very complex and outside the scope of this forum, so you would have to consult a competent Halachic authority, or in the parlance of the site, CYLOR (consult your local Orthodox rabbi).