The Lady J - probably a NSFW search but it’s been on the market for years.
Yes, it specifically refers to a series of rest areas / travel plazas on the Illinois Tollway system. They have gas stations, rest rooms, restaurants, and travel information.
Not the way they set up most of the Oases on the Illinois Tollway (the DeKalb one is built differently). The primary building of the Oasis is on a bridge over the tollway; there are separate on-and-off ramps, parking lots, and gas stations on each side of the Oasis. So, for example, if I’m driving northbound on the Tri-State Tollway, and get off at the O’Hare Oasis, I park in the lot for northbound traffic (which is on the east side of the Oasis); when I go to leave, there’s only one exit from that parking lot, and it is an on-ramp back onto the northbound Tri-State.
You can walk through the Oasis building from one side to the other, but there’s no way to drive a vehicle across (the building takes up the entirety of the bridge). And, the fact that you can’t drive across was the gist of the OP’s complaint. 
Duh, that was on his Tracks compilation!
No, not all-out. It has to be at least a little in. 
To be fair, there is at least one oasis where there’s a vehicle bridge to take northbound traffic to the southbound side of the roadway, where the buildings and gas station were built. Even there there are barriers separating the north and southbound traffic so you can’t change direction.
I had to double-check with my husband about whether he posts here, since that’s something he’d say.
I only wished they had a turn-around. We stopped at one (Belvedere?) a few years back, and after eating got back on the road. About a mile from the oasis my wife says, “Where’s my purse?”
It was something like 15 miles east to the next exit, then 15 miles west back to the oasis. We asked the shops there about lost and found, but no purse. We then walked to the east-bound parking lot, and looked about 200 feet down the ramp to the tollway.
My wife apparently put her purse on the trunk of the car, fiddled with something in the back seat, and forgot about it when she got in. Fortunately, the purse was a white canvas-style, and there was some construction cones on the side, held down by white sand bags, which looked just like her purse. Otherwise, I think someone would have seen it, stopped and taken it.
We got back on the tollway west bound, drove another 10 miles to the next west exit, and then turned around to head east again. It added about an hour to our trip. But we found the purse!
This is my complaint. If you miss your exit you often take a 10 mile trip just to turn around when there’s a perfectly good bridge over the tollway RIGHT FUCKING THERE!
There’s a rest area (east coast term for an oasis) on the median of I-95 in Maryland. The median gets really really large and each direction of the highway can get to the rest area via a left-lane exit. There’s shared parking and there are signs to help drivers get onto the road in the direction they want and avoid going the wrong direction. There are no signs indicating that turning around is not allowed. The road does have tolls but there are no tickets - you pay the toll when and if you actually pass a booth. For example, there is a northbound booth an hour or two northeast of Baltimore - if you get on right before the booth, then get off right afterward, you pay the full toll. If you get off right before the booth, you pay nothing.