Don't forget off site for a backup reservation. And you can continue to check onsite.
Also, think out of the box. The Doubletree Suites on Hotel Blvd sleeps 5 and shares a free bus to the parks with the other Disney Springs hotels. The website Mousesavers.com has a special deal with the Holiday Inn Resort Orlando Suites Waterpark from $74 per night with free Waterpark fees and free parking. A 2 bedroom King Suite at the Holiday Inn is $505 with taxes for your dates if you take the special link from Mousesavers.
Since you're driving your car, all of the condos and timeshares open up for you. You can rent a 2-bedroom condo or timeshare for a full week for the same price as two nights in a Polynesian studio. I would look for a Saturday check in even though you won't be there until Sunday. The only downside with a condo/timeshare is that they are NOT refundable so you have to be 100% sure that's what you want to do.
I know it's off site and you'll pay for parking at the parks but your dining reservations and fast passes won't be changed. Offsite is really not bad at all and a lot of people gladly forgo the 24/7 Disney "magic" for more space and savings.
I would:
(1) Book a refundable reservation somewhere else. Offsite if need be. Make note of when you'll need to cancel.
(2) Review your condo/timeshare options. The Orlando board here is a good place to start.
(3) Continue to look elsewhere onsite if that's really where you want to be. I do have to tell you that there is only partial
DVC availability for your crew of 5. There are currently two non-consecutive days in a 1 BR at Old Key West. That's it.
(3) Contact the DVC owner by phone and by email. Tell them that if they can't get back to you to discuss your reservation by a certain date that you will have to find other accommodations. Be prepared for them to drop you like a hot potato but you're been nice and patient and they appear to have your phone number on ignore.
Continued good luck to you! I hope everything turns out OK.