Hydrangea has been dormant for a really long time. I really want to get it to grow tall this year. Finally there are some signs of growth from the root. Should I cut the dry branches?
Your hydrangea probably suffered either cold damage, root rot (too wet substrate) or the pot was too dry at some point of over wintering. Usually all ot takes to get hydrangeas to grow after over wintering is to give some warmth and moisture. If yours is reluctant to grow in spite of being in a warm conditions and properly watered, then it could have had some serious issues with watering. Check the roots. My neighbor is growing many hydrangeas and she tries to postpone any pruning if possible, as hydrangeas will bloom less after strong pruning. Only if its absolutely clear, that the damaged branch will not wake up, she removes it. Hydrangeas are usually vigorous and recover damaged vegetative parts quickly.
I think your plant would do better in either a much larger pot or in the ground. Hydrangeas are usually very easy plants in this area and should not need much attention aside from regular watering and occasional pruning.
Just saw this thread, Hi Alex. Cut everything down to the new shoots now. It will recover, but all the wood in the photos is dead. I would consider a repot also. I think the compost/ soil in that pot is spent. It probably won't flower this year if it is the mop head variety. But it might if it is the paniculata variety. TBH though, I would be looking at 2023 for a full recovery.