Mr. Lieberman: the asker had indicated that he was employing the ITRANS scheme wherein the "z" does correspond to "ś". As for the speculations on the second component, it is not from dhi or dhA but it is separate free word "ADhya" which means rich, opulent etc. and Monier Williams dictionary says it is at least as early as late Vedic texts such as zatapathabrAhmaNa. Anyhow that word is pretty commonly compounded with others such as guNa, gandha to give guNADhya 'endowed with good qualities', gandhADhya 'rich in fragrance" etc. ADhya's retrfolex is speculated by the same dictionary from some original form with -rdh- sequence changed to -Dh- due to some prakritism.

The study of Sanskrit is something I wish to continue all my life. It is a wonderful and beautiful language, which gives up its treasures only reluctantly, and with time. I do not wish to pose as an expert, or, as being particularly knowledgeable. Simply, as an enthusiast, who might be able to help with a particular question, depending on how long it has been since I have been involved with the language. As far as I know, there is no "z" in Sanskrit, except for those words borrowed from the Persian. No doubt, the z you use, is a convention used by a scholar I don't happen to be familiar with. In Monier Williams Sanskrit-English dictionary, the Devangari letter "श" is transliterated as "ś". In Sanskrit, पाश pAśa is, as you suggest, a snare, trap, noose. An epithet of Varuṇa, is पाशधर pAśa-dhara -- holding a noose. We find an occurence of "pashadhya" in the second strota of the Sahasranama: उद्यद्भानु सहल्राभा चतुबहु समान्विता रागस्वरुप पाशाढ्या क्रोधाकाराङ्कुशेजऴला - २ 2.Udyadbhanu sahasrabha chaturbahu samanvita
Raagasvarupa pashadhya krodha karankushojjvala. The subject of how letters become transmuted into other letters is a basic question which I have found on line, but, alas, do not know by heart. There are rules for turning "weak" vowels into "stronger" ones, and, there are cases where a change in a consonant follows a change in the vowel. I apologize, but this is the best I can provide, in terms of answering your specific question.