If I got this right,
1) You paid a $75 deductible to be told the dishwasher was unrepairable and should be valued at $500 for replacement
2) Select Home Warranty insisted on a second opinion and made you pay another $60 - same verdict. The contractor they sent out is their responsibility, not yours.
3) SHW offered you only a small depreciated value for your dishwasher, $125, instead of the replacement cost implied in its pamphlet and is also telling you that it's older than it actually is as part of the rationalization for the low value.
Here are my suggestions:
a) Assuming you didn't literally pay cash for your dishwasher, you may well be able to find a receipt. Contact your bank, credit card, manufacturer and/or the appliance store. It may take you some time to get to the person that can help you, so persevere. In a shining moment for both Sears and a manufacturer, Sears found my original receipt for a fridge that was several years old and the manufacturer repaid me the full original cost. It did take patience. The manufacturer may be able to track the appliance, too.
b) Go ahead and buy your new dishwasher, because you're not going to get any further right now and you need it, right? Submit your invoice for the dishwasher and the extra deductible SHW owes you. If they send you a check for $125, either don't cash it or make sure you write somewhere "encashment does not constitute acceptance as payment in full." Gather your evidence - the brochure, your contract, the receipt from the repair company (which probably stopped answering your calls because they could do nothing else to help you), associated emails, and your receipts.
c) Send a detailed formal complaint to SHW's General Counsel by certified mail.
d) If there's no joy there, depending on what state you're in and whether they have a presence there, you may be able to file in Small Claims. The Clerk of Court may be able to help you figure it out although they can't give legal advice. If you have a strong enough case but would need to file in NJ, you may also be able to travel to NJ and add the costs to your case. Explore all this before you finalize your letter to SHW. If what you say is correct, I'd be ticked off enough to file where I had to and not worry about the cost, but that's me!
From what you have written, SHW seems to have been deceptive throughout - first by implying that you'd get a replacement in their advertising and then backing off in the fine print, if that's what they did - then telling you you wouldn't have to pay a deductible - then misstating the age of the dishwasher. Unfortunately this is a common problem with warranty companies, especially those offering low premiums.
"Unreasonable delay" is also a thing. People by homeowner warranties with the expectation that they will get immediate help with a problem.
Personally I would forget about homeowner warranties and just put the premiums in a designated account, but of course that doesn't help with the immediate issue.