I will forgo a detailed technical explanation which is better suited to Linguistics SE (the reason has to do with the three clauses and "who is" – they all have to refer to the same person). In fact, under rules of English, it is impossible to construe "is not employed on a full-time basis" as referring to "you" (Eugene), and the clause can only refer to the "children". A non-technical common-sense way to see this comes from substituting your name in here and shortening the verbiage
Dependent means Eugene and your children, or a child for whom you or
Eugene are the legal guardian. And that child must be under the
termination age, not be full-time employed, and must be dependent
on you or Eugene for support.
The idea that Eugene must be dependent on Eugene for support is a bit nonsensical and contradicts the plainer fact that Eugene is identified at the very start as being a dependent.
There is a possible ambiguity regarding the scope of the "not-employed (etc.)" clause. The sentence identifies two kinds of "children": natural, adopted or step-children, and also a child that you are a legal guardian of. The question is whether the clause "who is under the termination age, is not employed on a full-time basis and is dependent on you or your spouse for support" refers to all of those kinds of dependents, or just the legal guardian type. There is an interpretive doctrine, the last antecedent rule, which would say that the restrictor only applies to "or a child for whom you or your spouse are the legal guardian". However, that rule always gives way to "common sense" and "the full meaning and intent of the document", so in fact it is typically only invoked as a way of rationalizing a conclusion reached on other grounds.