Hello, ohiokirsten. Old plants that bloom poorly can be pruned to renew themselves. Before doing this though, consider doing a soil check if for no other reason to rule out other types of problems. Once flower buds develop in the fall, failure to open, indicates improper pruning after bud set, a pest problem, some type of mineral deficiency (or excess) as well as moisture issues (too much or too little).

For example, if you have a dry & warm winter, the plant will break dormancy and start needing moisture. If conditions change and get colder, the plant will try to resume dormancy. Under those fluctuating conditions, the ground would not be frozen during the warm spells so the plant would be looking for moisture; if this happens, I would try some watering (when possible) in case the plant awakens from dormancy. Here in Texas, the ground never freezes so I always water about 1g of water a week per plant during winter.

The amount of sun light that any plant gets will help develop flower buds. All things being equal, the more sun, the more flowers the plant should produce. So try relocating the plant if the soil test uncovers no problems. While I have no choice but to find a spot that gets shade by 10-11am, you should be able to give them one that is sunnier... say shade starting at 3pm-ish? Tip: observe in your area others who grow hydrangeas and determine when do their plants receive shade in the afternoon. Some people in the northeast and Chicago have them in full sun conditions (> 6 hrs of sun)!

However, even if you transplant to this new location, you will still need to address the deer problem.

And yes, those "volunteers" that are growing besides the original bush can be transplanted. That is a propagation method called ground layering. Once they develop good roots, sever the link between the mother plant and the "volunteer", leave it in its current spot for about a month (monitor its health) and theeeen transplant it into another spot or into a pot/container.

Good luck ohiokirsten! Post pictures of their progress if you can,
Luis