Unavailing rhymes with:
prevailing, surveilling, unveiling, veiling, ailing, assailing, ayling, bailing, baling, curtailing, derailing, detailing, emailing, entailing, failing, flailing, grayling, hailing, impaling, inhaling, jailing, mailing, nailing, prevailing, railing, regaling, sailing, saling, scaling, surveilling, tailing, trailing, unfailing, unveiling, veiling, wailing, whaling
Unavailing sounds like:
unappealing, unbalance, unfailing, unfeeling, unveiling