I have a 97 Ranger Sport R70 with the same system as you, so I'm very familiar with it.
First off, When in empty mode, your valves are open. I don't believe the older models like ours have a check valve in them to prevent water from entering the livewell like the new ones, and if they do, they are simply worn with almost 18 years of wear on them. With the valves we have there will be a minimal amount of water in the livewell with them in the empty mode. They won't however be completely full, and if you check after a little run down the lake they will be emptied out until the boat settles back in the water. Once you flip the valve to fill, a flap is slid into place over the drain hole in the valve. It allows water to flow in when the pump turns on, and when the pump turns off, the pressure from the water in the system presses the valve back against the valve body sealing the system. Once full the system slowly pulls in fresh water while recirculating the water currently in the livewell and the extra flows out via the over flow hole.
As for the one livewell not draining, there are a few possible culprits. Either the drain is plugged, or there's a valve issue. If the drain is plugged, I've found the best solution to be to take off the screens on both in the livewell (if you have them...if not you should) and under the step. Take a garden hose with preferably one with multiple spray patterns and put it on the jet setting and spray from both ends. Take your time and go back and forth with it and usually that'll clean them out. It could be the valve itself as well. If the arm broke, or the insides no longer move that could cause the same issue. It's also possible that the cable that runs from the controls under the throttle to the valve could be stretched or broken. The only way to check that is by getting to your valves. They're located under in the rear compartment under the false floor in the middle. You may have something over the false floor such as an oil tank, or battery, but if you move that you'll see there's 4 screws that hold it in place. Take them out and the pumps and valves will be out in the open.
If you need more information, or visualization of the system, much more info can be found at https://flow-rite.com/marine Our Rangers run the system #2. All parts are still available, and should be direct replacements if you need to replace anything. Also the valves can usually be rebuilt as well if that needs to be done.
Hopefully this helps, if you need anything else, just let me know.