I have a few little giants and they can be a challenge. It sounds like the sensor went bad in yours. I had that happen and they replaced it when I called customer service. They also sell a replacement for the digital controller and a new sensor for about $35. The humidity sensor doesn't ever change unless it's above 50%, it just shows 48. You probably already have a seperate humidity sensor, if not you can find a wireless one that uses Bluetooth and links to an Android app to record humidity and temperature for about $5 on ebay. They do take about 3 weeks to arrive though.
I wouldn't recommend oiling the membrane. It reduces porisity and it doesn't rehydrate the membrane. When it's yellow it's still flexible, but harder for the bird to tear. Brown or tan and it's dry and will either be papery or leathery depending on how wet the egg is. With your egg I would have torn away the dry membrane without enlarging the hole so that you could see the bill.
I'm glad it worked out well.