would a 80 watt adaptor charge faster than a 60 watt

I would not use OEM. Buy a new adapter. The only advantage of using the 85 watt charger is the capability of providing power for more peripherals and to stop the need for battery power when the computer draws more power than the smaller adapter can provide. I may allow for slightly faster charging depending upon the battery software which may restrict current to the battery in order to prevent overcharging.

You can safely use a higher wattage adapter, but your Mac will not charge faster. The computer will draw only the power it requires.

¿Only the DC cord is damaged?

The DC charge cord on a USB-C Power adapter is replaceable by itself, it is Not captive in the Power adapter.

Buy Apple Brand, or be sure you get one rated for the higher Watts.

Also, as I was looking these up, I noticed a new Power Adapter does NOT include a new charge cable.

@Kappy, seems that you are receommending the OP use a 3rd party (non-Apple) power adapter... In general usage, the opposite of OEM is aftermarket or 3rd party, i.e. Anker, etc.

OEM = original equipment manufacturer. Technically, the OEM for the 87 W adapter with the Apple logo on it that's sitting on my desk is Delta Electronics (a Tiawanese manufacturer, although the factory that produced the adapter in in Jiangsu, China). They make them for Apple to Apple's specifications, and while you can get a Delta Electronics power adapter that's not Apple-branded (e.g. https://www.amazon.com/Delta-Electronics-Notebook-Touchbar-Tip-Only/dp/B07HYTSWT6), they're not made to Apple's spec.

Read, again, my friend: "I would not use OEM. Buy a new adapter." OEM does not mean it came from Apple. I specifically suggested the OP buy a new adapter, meaning one from Apple. I think you misunderstood my reply.

OEM is one of those terms that just causes problems.

"original" in this use is supposed to mean "novel, unique"

a third-party company that puts together parts from disparate suppliers to produce a novel product is called an Original Equipment Manufacturer, OEM.

on its face, the term sounds like it should mean the original, as in straight-form-the-source of the product.

.. and there lies the confusion.

Well, I could buy a brand new power adapter from Anker and meet the two criteria you listed: not OEM and new.

While the term is technically problematic, in this context the adapter from Apple has an Apple logo on it just the computer. You can’t buy an iPhone from Foxconn, so for all intents and purposes Apple is the OEM. Thus my issue with your recommendation not to buy an OEM.

OEM = Original Equipment Manufacturer = Apple in this case.

At least that's my understanding.

https://en.wikipedia.org/wiki/Original_equipment_manufacturer