Presta to Schrader valve adapters that don't screw on the valve core

Never seen one myself - there are pumps that have push-on "speed" adapters for presta, but they are part of the pump head and you wouldn't find one at the service station.

The best fix is to unscrew your valve core and add a small dab of hand-strength threadlocker. The Locktite one is normally blue, but other brands have other colours. Then reinstall the core and snug lightly. You can reinflate immediately.
Once this is done, there's much less chance of the core coming out with the thread-on schrader adapter which you still need.

Another possible solution is to simply carry your own pump and not depend on machines. A quality presta capable minipump is cheap enough, and will be with you if the puncture is far from civilisation and fuel stations. Minipumps are slow and tiring, so you might choose a "small floor pump" which are faster but larger, while remaining smaller than workshop-sized floor pumps. I have one and its somewhere between the two for speed. Or you could carry CO₂ sized for your presta tube.

You should always carry the small threaded schrader adapter too, along wiht the usual on-bike tools and emergency spares like master link and tyre levers.

SKS makes a press-on adapter. Unscrew the valve nut, push the adapter on, use the schraeder head on the air hose, pull the adapter off and thread the valve nut back down. Tell your shop it's p/n PU4001 from Q.