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I would put it off until fall if at all possible, or wait until next spring. However, as long as the plant is properly acclimated beforehand and the temps are not too hot/cold, you can still go ahead and move your hydrangea now. Be sure to place it in a suitable location and give it plenty of water deeply until the plant re-establishes itself. Also, make sure to get up as much of the roots and surrounding soil with it. You may want to prune it back some too so that the plant focuses on root growth rather than foliage/blooms. Here is information on their care: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/growing-hydrangeas-hydrangea-care-guide.htm

You may be lacking phosphorus. If there is a lack of phosphorus in the soil, plants will have little to no blooms. You can either use a phosphorus rich fertilizer or add some bone meal to the soil. These articles may help as well: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm

It is likely lacking a good phosphorus/nitrogen balance in the soil. Too much nitrogen or too little phosphorus will cause it to fail to bloom. This article will have some other suggestions as well:
https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/hydrangea-not-blooming.htm

Judging from the symptoms, I would blame underwatering. I know you said that they get watered daily and are in a good mix of sun/shade, but if you are having an unusually hot summer or if the hydrangea is getting more sun than in the past, this can cause problems. Sun amount can change if trees or buildings around the plant get larger or are removed. Even the water needs of the plant can change as it gets larger, which means what may have been enough water a few years ago is just too little now. Try increasing the water.

You may also have a root issue that is preventing the plant from taking up enough water. It may be mild root rot or bugs. Check the roots and make sure they are healthy.

The plant may also be having trouble taking up enough nutrients, so you may want to fertilize as well if you have not done so this year yet. Make sure you water well before you fertilize to keep from burning the roots.

It is common for the flowers to become heavy and lean or fall.

Here are links for you with more help.

https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/prune-hydrangea-bushes-hydrangea-pruning-instructions.htm
https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/growing-hydrangeas-hydrangea-care-guide.htm

No leaves on the plant is likely an indication of pests or disease.
Treat with Neem Oil as it works as both an insecticide and fungicide.

https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm