K.,
Bedwetting runs in my family so I am very familiar with it. Pull-ups/Goodnights are probably fine while you are making sure he doesn't have a medical problem. But they do have some drawbacks. Since they won't let him feel that he is wet, they probably won't encourage him to stay dry. In kids that have never been dry, the first thing you do is take them out of the Pull-ups/Goodnights. There are washable pads that you can put under the child on the bed, but they are expensive. Sometimes we used incontinence pads taped down with masking tape while my son was learning to stay dry. They are even more expensive in the long run, but you aren't committing quite as much money up front. They are available in most (all?) drugstores.
Before you do anything though, I suggest you check with your doctor. Bedwetting in a child that has been previously dry can be a symptom of medical problems like UTIs and other things, some a bit more serious. If it is a UTI, antibiotics could clear it up fairly quickly and maybe he will be able to say dry at night fairly quickly. I'm not a doctor but I know from experience that similar symptoms (bedwetting and daytime urgency) in a small child can also be the early signs of chronic constipation and that needs to be treated as well.
I think I would also check in with him, if I were you. Is he afraid to get up at night? Would a night light help? You said you moved 2 months ago, so it seems reasonable that he might still be adjusting a little.
Good-luck,
K. L.