The CMD community is one of avid home and a few professional bakers so hopefully others will chime in as you ask questions. I also advise strongly to search the board archives as their is a wealth of information stored in the many threads over the years.
Regarding your ingredient question, many on this board use mixes with pudding in the mix all the time and still add additional instant pudding mix and the other ingredients called for in a recipe. It would be very helpful to know your location in the country
as higher altitudes and also humidity play a role in a cake's texture, moisture level, etc., Also, do you have an oven thermometer?
You can pick up a reliable yet inexpensive one at a kitchen supply store or possibly a Michael's in the section with all the baking supplies. You would be surprised how "off" an oven's temperature can be from what is stated on the dial and if you're not baking at the proper temperature that will have an effect as well. Also, some here have found that their baking times are always longer than those stated in the book so the baking time is only just a guide.
Unfortunately I have not attempted the specific cake you tried so I cannot even hazard a guess as to the over moistness issue but likely somewhere here can offer suggestions. Finally, as far as your quest for cake mixes with no pudding in the mix I would try looking for the store brand / generic mixes in your local markets as those, like Duncan Hines, are the non-pudding mix varieties.
Welcome once again & Happy Baking!