Sticky gums are a sign that your horse is dehydrated, so make sure he has access to clean, fresh water at all times. If you struggle to get him to drink when at a show, try taking bottles of water from home. Some horses are very fussy about what water they drink, and can taste very subtle differences in the mineral content of the water, which deters them from drinking. If this doesn’t work you can put additives (like apple flavouring) in his water to encourage him to drink. There are several of these on the market currently.
NOTE: Examining the gums for stickiness and testing your horse’s capillary refill time is a much more accurate way of measuring dehydration than the skin pinch test. The skin pinch test is only positive in VERY dehydrated horses, whereas the other two will pick of milder levels of dehydration.