13 is quite old (for most breeds).
I would get him checked over by a vet, the shaking could be caused by many things. He could be just getting a little more clingy because of his age - remember if you go by the rule of 1 of our years is equal to 7 of their years then your dog is now around 91 years old!!!
It would be well worth getting him a general check over by a vet, they could well pick up on something early and may be able to help him have a better quality of life for his remaining time with you.