Dogsailors... many cruisers are out there with dogs. I knew one cruiser that had 5 of them (tiny yorkies). I can understand not wanting to leave your children behind but as noted above it is quite problematic in many places...and may not be in the best interest of the dogs health. Nevertheless, many cruisers do it.
Given your plans, I assume you are east coast based and you can certainly day hop there giving the pups a place to get off the boat and adjust to it.
The Bahamas can offer many months of cruising and are not too restrictive on dog entry (advance permit required) and again...offer land readily available for walking/exercise. You can judge from there how they will do on extended passages and may wish you modify your cruising plans accordingly. There are some tougher places in the Caribe to enter with pets...(BVI's are one as I recall) and sometimes you will need permits in advance of your visit so you need to be sure your dogs can withstand multiday passages at sea and full time living aboard and seasickness issues before going much further than the Bahamas.
Right now the best thing you can do is drop the hook somewhere and stay just on the boat with them for a few days and see how you both do. No trips ashore in the dinghy allowed. If you're all still happy after 4 or 5 days...maybe your plans are not unrealistic.