Sounds like it's going to be a hard summer for you, too.
And, since you are sure to be under a lot of stress, some of that will rub off on your son. The main way I know to smooth the transition is to link the new with the old in as many ways as possible.
Drive by and see the old house and talk about things you did there. Take him to visit the dogs, wherever you have them kenneled during the moves. And, unless you're working full time in addition to managing all these moves, I wouldn't enroll him in any daycare for a while.
Perhaps you can find a playgroup near your interim apartment where he can go two or three mornings a week. If so, he will have contacts with other children but not be away from you for long periods. In fact, if he cries when you take him, I'd stay with him. And his prized possessions are important. Make certain that his favorite toys can easily be found among the sea of moving boxes and unpack them first. They will serve as a link to his earlier home and help him move on toward the future.
If it gets too much for him, you will probably note it in disturbance of routines—he may resist going to sleep by himself in the new place, may show less interest in his food, may regress in his toilet habits. Be tolerant if these signs appear and give him some extra love and support. Only if they become very severe and last a long time would I seek any outside help.
Chances are your son will handle the transitions just fine if you and your husband do.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.