All the tips given should help to focus on your technique, which could be a reasonable question to bear in mind each time you face a similar problem, but I would focus on the key fact that you have ensured this problem only takes place using that guitar.
First of all, as you suggested, replacing strings would probably help. Maybe it's not a solution by itself, but a damaged or too used set of strings could produce not desired effects, and replacing it should be a usual maintenance action for guitars. If you are able to, check the exact model and gauge with the rest of your guitars. Sometimes a string model do not fit for you due to it's winding and construction material, and depending on your way of playing (as very often is recommended, a teacher may help on this matters...), but specially on your fingers touch and feel. Please, if required, take a look at any related web resources, as nowadays there're plenty of options to test (e.g., https://www.musiciansfriend.com/thehub/how-to-choose-the-right-strings-for-your-electric-guitar)
You refered that the guitar is configured with medium action, have you compared this with any other guitar you have? Changing action do no have costs associated if you are familiar with it (otherwise, you might talk with a luthier to make the experiment), so you could try readjusting this guitar and give a try.
Each one of us have a different hand/finger structure, and this might have an impact in playing action, depending on the guitar configuration and adjust, the main goal should be removing differences between configurations to determine if any of them is causing this undesirable effect.
Moreover, using the right amount of hand or similar product may not be a final solution, but, if this works, I would recommend to keep using it. As mentioned before, our hand may differ not only in structure, but in skin. Thi tip usually helps to get over the hand dryness on people with certain skin affections.