I know what you mean about those little mushroom heads. I never found them for sale except when buying a whole new socket. Try an automotive store for that. It is also possible Walmart's automotive department has them. If all you want is the mushroom, use the new one with pigtail and discard the rest of the new socket.
Consider converting the fixture (and all the others inside) to LED for more light and also so your house battery lasts a lot longer. You can buy replacement LED bulbs that fit, or instead use LED strips like I used converting all my RV florescent fixtures
HERE.
Those LED reels are great. At home I installed a continuous strip under the kitchen cabinets all around the counter. I mounted a low voltage on/off dimmer controller in a convenient location under a hanging cabinet and placed the 12V transformer inside the rarely-used cabinet above the stove, utilizing the outlet that powers the microwave hood.