Now, when dormant, is the best time to move them. But you can also move them in early spring or in the fall... as long as the soil has not frozen.
Big Leaf Hydrangeas prefer morning sun until 11am or earlier. They can also be sited in a location that receives as little as a couple of hours of direct sun but, they will bloom best with a few more hours of morning sun. They only tolerate full summer sun in high elevation areas or in coastal regions whose summers are very mild. Summer temperatures in full sun should rarely reach or exceed 85°F. Think of Cape Cod, MA for examples. Depending on your summers, Big Leaf Hydrangeas could also encroach into a few early afternoon hours provided that your summer temperatures are not extremely harsh and you provide supplemental watering in dry/hot periods. You will need to test the shrubs in those spots to confirm they can take (and how much) early afternoon summer sun.
To find out how much early afternoon summer sun they can tolerate, grow them in a pot in the location being considered for a growing season and see how they perform. You can place the container on the ground or you can dig a hole for the container and place the container in the hole. If at any time the blooms and foliage suffer from too much sun, immediately move the pot to a shadier location. Once they perform to your satisfaction, you can plant them in the fall or once temperatures typically stay under 85°F.
You could try the spot that gets less summer sun (3-4 hours) in this manner. It will depend on how strong the sun is there during those 3-4 hours as the leaves and the blooms cannot tolerate much direct exposure to harsh summer sun. Provide temporary shade in an emergency using outside chairs or outside umbrellas.
Moving them back a few feet closer to the evergreens will provide the basic sun exposure they prefer. But, you may have to negotiate with your neighbor a reduction in the excessive amount of watering or consult with a landscape company to see what do they suggest in order to drain the excess water away. Sustained soggy water conditions can cause root rot in hydrangeas.
Ensure that the shrubs are watered consistently and sufficiently. Water the soil -never the leaves- as soon as the soil gets dry at a depth of 4 inches, the typical depth of new hydrangea roots. To do that, insert a finger at that depth often in several spots under the canopy during the hot summer months or in dry periods. Do this almost daily in the mornings. To ensure you are providing sufficient water, after you water, periodically wait for the soil to percolate and then insert a finger to a depth of 8" to ensure that the soil is moist (if dry, use more water; if soggy, water less). Also ensure you also have 3-4" of organic mulch past the drip line at all times of the year.
Being compact, these Can Do! lacecap hydrangeas can also be grown in pots. Not sure if the pots could be sited in the soggy soil effectively without issues though. Maybe in other shady places? You can also plant equally compact panicle hydrangeas where these Big Leaf Hydrangeas are currently located. Panicle hydrangeas currently include very compact varieties and they are more tolerant of summer sun in NJ, even from sunrise to sundown (but not everywhere; not in full sun in the southwest with its extremely hot temperatures). The only problem is that 3-4 hours of sun does not promote lots of blooms in panicles.
