Hello Jane! The "oily" effect I see now is simple separation; this is part of the liquid component of the paint which separates out. The normal way to get around this is to ensure that the cap is securely screwed onto the tube, and then gently squeeze the tube all over, and this tends to mix the liquor into the paint.
I can't answer the stay wet palette, with your paints going a little mouldy, but I can only offer the simple advice to use less on your palette, so that you don't need to store the paint so long, and have less wastage.
I hope this helps,
RSC Wigan / Warrington