I've got to ask the question... what is a malipoo?
Okay, couple of things....
Yes, it is not uncommon for an adult dog to be frightened of a new puppy. Puppies are very bitey with very sharp teeth & they are often very hyper. This can equate to utter misery for an older dog unless he can be provided with relief from the puppy.
It is very important that you provide a means of escape for your older dog. This can be achieved often by simply purchasing a fairly low x-pen, opening it out & using it as a room divider. You can then perhaps put a chair or something nearby that your older dog can use to jump from one side to the other. This will give him the option of either playing with the puppy or having a break from him.
DJ runs to me for help so I pick up the puppy to come DJ down.
By all means, pick up your puppy for cuddles... who could resist that? Certainly not me! However, be aware that height plays a very important role in the pecking order of dogs. A dog that is place in a physically higher position that another dog is being placed in a more alpha position than the other dog, and this can cause problems. I would suggest that when the puppy is being annoying, you just hold him back with your hands, but do not pick him up. Gently push him away & into an area where he can not annoy your adult dog.
I know they need to work it out, but I don't want the puppy hurting DJ or making him more timid than what he is.
Yes, in the end, they do need to work it out, but you can make the transition a little more smooth, and along the way start setting some ground rules for the puppy.
Two is great fun, three is even better... so have a great time.