The punctuation of an adjective clause depends on whether it is essential (restrictive) or nonessential (nonrestrictive) and uses commas accordingly.
An adjective clause is essential if the information it contains is necessary to the meaning of the sentence: The word "vegetables" is non-specific.
However, if the additional information does not help to identify more narrowly the identity of the noun antecedent but rather simply provides further information about it, the adjective clause is nonrestrictive and so requires commas (or a spoken pause) to separate it from the rest of the sentence: Depending on context, a particular noun could be modified by either a restrictive or nonrestrictive adjective clause.
The connecting word "that", if not explicitly included, is understood to implicitly precede "I won" and in either case functions as a subordinating conjunction.
This sentence also includes two independent clauses, "I cried" and "I didn't faint", connected by the coordinating conjunction "but".
Dependent clauses may be headed by an infinitive, gerund, or other non-finite verb form, which in linguistics is called deranked.