Cardboard boxes are fine if all they do is shred them up. You just don't want to used the ones with glossy paper, as they have plastic added to them. If your bun chews and eats more than a few nibbles, then it might not be the best thing, as too much ingested paper and cardboard can possibly lead to digestive impactions.
Though eating pellets, veggies, and hay does cause adequate tooth wear for most buns, some buns with special tooth problems may need a little additional help with different things to aid in tooth wear. Honestly your buns incisors aren't bad at all, and the front teeth are probably just fine. But if you want to add things in to help with keeping them worn down, you want to use things that your bun will specifically have to use his incisors for. Incisors are used for chopping and cutting up things. They will use them somewhat for grabbing and chopping up their pellets, veggies, and hay, but the things that buns use their incisors most for are stripping the bark off of apple branches and certain other woods, chewing on larger pieces of wood, hard dried pine cones, and also with hay blocks to chew the pieces off. So these are the types of things that would help with incisor wear.
You don't want to just use any odd stick in the yard, as some wood isn't safe for rabbits. You need to make sure you are using a rabbit safe wood and something that is clean and doesn't have mold, moss, or plant rust, and hasn't been sprayed with pesticides. Apple and willow are probably the most common ones used. You can also use the wood block bird toys as long as they aren't the ones with string and rope, as you don't want your bun chewing and ingesting those. You can also use plain pine blocks that have been thoroughly dried. I buy 1x2's at the hardware store and chop them up for my buns to chew on if they want.
http://www.medirabbit.com/EN/GI_diseases/Food/Branch/Branch_en.htm
If you were using alfalfa hay blocks, that may be why your bun gained some weight on them. If you can find ones that are just plain timothy, then it would be better to use, as alfalfa will have too much protein and calcium for most buns. Using a timothy cube isn't much different than giving your bun timothy hay, just doesn't have the long stems that loose hay has and which is generally considered the best for buns. It's just timothy chopped and compressed into a cube. So though cubes shouldn't replace loose hay in your rabbits diet, they can be a good addition to the diet for buns that like to chew on them.
The incisors wouldn't be likely to cause any drooling. It's more likely to be due to molar spurs, though there are other conditions that can cause drooling as well.
http://www.medirabbit.com/EN/Dental_diseases/Differential/excessive_en.htm
Its really wet and windy today is all!
