The agent in ME told me that we can just go to the ME booth in the morning and they will be able to give us vouchers to ride to WDW.
I assume you were talking to a room reservations agent, not to a DME Guest Services agent. The agent gave you bad information. Disney's contract with Orlando International Airport requires all DME passengers to have advance reservations for DME.
You can -- and must -- make an advance reservation for DME. Your plan is to take DME to your WDW resort on your resort check-in day. Only you'll be arriving from the Hyatt elevator instead of from an airline arrival date. That's perfectly fine.
Call Disney's Magical Express Guest Services at 1-866-599-0951. Tell them the approximate time you want to depart from the airport on a DME motorcoach. Don't be surprised if you receive DME vouchers with a flight time of 0:00.
You will not be able to use the DME baggage service for free baggage delivery to your room. Instead, the driver will put your bags in the baggage hold of the motorcoach. When he returns them to you at your resort, a tip is customary. If you get help from bell services at your WDW resort, another tip is customary. Because you're arriving at your resort in the morning, there probably won't be a room ready for you. Even if bell services just stores your bags until your room is ready, a tip is customary.
Then on the way back we should call and schedule the ME 24 hours before we need to be picked up. Since we do not have flight numbers for those days they can not schedule the transportation ahead of time for us. She said we have to be staying at a WDW resort on the day of travel on ME.
There's no reason to wait until you get to WDW. Just tell Disney's Magical Express Guest Services the approximate time you want to depart from your resort on your check-out day. Explain that you are not flying that day. There are typically two pick-ups per hour at each WDW resort, if guests need them. You will not be able to use Resort Airline Check-in (RAC) because you're not flying that day.
I thought we were being smart not paying Disney prices and wasting ticketed days by doing this. Now I am having doubts that we have saved any money if we can't ride on the ME.
You are being smart. You can still use DME, just not the inbound baggage service on your check-in day or RAC on your check-out day.