I like to rent timeshare condos from owners if I am doing a Fri-Fri, Sat-Sat or Sun-Sun stay for a great deal on sites like Redweek.com or TUG2.net classified ads section, and FYI (uber/lyft is really easy). If it is a shorter stay, I often pay a lot more per night and do retail rentals at the same places or some place like a Residence Inn, but quality is still great and you get so much more for your money than onsite Disney.
Here are some places that I have stayed and liked:
Closer to Disney, but it's only a 1/2 hour to get to Universal
-- Cypress Pointe Resort and Cypress Point Grand Villas (close to Downtown Disney and tons of restaurants, etc) -- they have lots of three bedroom/three bath condos. You can even walk to some food options. If you do this you might want to do the three bedrooms with the two masters and the two bedroom has a pretty small bedroom with just a double bed. There is a pull out bed too, though. You know what would work for you.
--Sheraton Vistana Resort (down the road from Cypress Pointe) no any restaurants you can walk too, but some food options onsite (very nice two bedroom/two bath condos). Lots of activities -- They have little man made ponds so your patio or balcony will usually have a water view. You could have the master. Your 22 year olds could have the other spacious bedroom (two doubles).
Very close to Universal -- 1/2 hour to get to Disney
The Sunshine Resort -- condo itself super nice and a fabulous huge balcony (incredible). Didn't like the surrounding area as much. Had a thrill park you could walk to. Similar setup to the Vistana with bedrooms.
Between Disney and Universal (by SeaWorld) -- We usually do SeaWorld, so like staying in this area. There are not a lot of restaurant options as close, but your distance between Universal and Disney is pretty equidistant.
--Hilton Grand Vacation Club SeaWorld - great timeshare two bedroom condos (walking distance to SeaWorld, which I realize is not on your list)
-- Of if you prefer a hotel, we on short stays have liked the Marriott Residence Inn SeaWorld (You can get two bedrooms). This has a king bedroom and a double queen bedroom and common area with mini kitchen. It's not as spacious and doesn't have a balcony or patio like the timeshares, but includes breakfast every day, has a restaurant there, etc. and fine pool area with hot tub. Grounds aren't as stunning or spacious as the timeshares, but this is a nice place.
These are the properties we picked on our last five Orlando trips. We like multiple bedrooms, a kitchen, space to spread out and can recommend all of these.
Here's one I haven't done before. It is a little more out of the way, so probably not as good a choice for you. We are getting a three bedroom/three bath here though for two nights in February (doing one day at the Magic Kingdom for about eight hours and having Orlando relatives come over to our condo and will order pizza and salad for dinner). After the mini stay we are doing a cruise on one of the big RCCL ships out of Port Canaveral. Anyway, to change things up are staying at this one: Grand Beach Phase I
Uber and Lyft are easy to do in Orlando and our next super mini stay we are doing that vs. renting a car. Renting a care is usually less expensive, but uber / lyft is not too bad to budget if you don't like to drive in Orlando (I like driving there less and less these days, so am doing uber and lyft a lot more).