I’d also suggest applesauce. Replace at least half the oil with applesauce. Also, if the cake calls for all white sugar, replace half of it with brown sugar.
Make sure you are measuring the dry ingredients properly, especially the flour. Spoon the flour loosely into the measuring cup, then level it off with the back edge of a knife. Don’t just use the measuring cup to scoop flour of the bag or the flour is too packed and you end up using too much.
When baking, set the kitchen timer for at least a few minutes less than the suggested baking time to make sure you don’t overbake. You can also get an inexpensive oven thermometer to check that your oven is heating properly. My last oven was always 25 degrees too hot and I had to adjust the oven temp accordingly. If you’re using a pan size other than the one called for in the recipe, you probably have to adjust the cooking time as well. For carrot cake in particular, a recipe that calls for canned pineapple tends to be more moist.
Finally, try to find a recipe that’s been reviewed by other bakers. I like www.allrecipes.com because I can look for a recipe that’s highly rated, read the reviews, and see what changes worked best for other people before I try it myself.