Azaleas are rhododendrons so such generalities are not very helpful. Some rhododendrons flower without much light and some azaleas do also. It varies by variety.

Regarding water, both rhododendrons and azaleas are vulnerable to root rot and should not be kept too wet. They need well drained soil. Again some varieties are more vulnerable to root rot, but that includes some rhododendrons and some azaleas.

Your comment on leaf burn from fertilizer bothers me since too much nitrogen will keep rhododendrons and azaleas from blooming.

Here are factors that may reduce the number of flower buds:

Pruning. The buds are formed in summer and early fall so pruning then or later is not advisable since it will remove flower buds. New leaf buds will form in the spring, but new flower buds won't form until the next year.

Variety. Some plants will never bloom. Some rhododendrons that come from the seed of a hybrid plant will look good but will never produce flowers or will produce very poor flowers. To come true to the parent plant, a hybrid may be propagated by cuttings or tissue culture but not from seed. A good hybrid seedling only comes about once in a while. For that reason it is important to know that you are getting a good named variety or a good species.

Fertilizing. Nitrogen promotes leaf and branch growth and discourages flower bud production. It can also force late season growth that gets killed or stunted by frost damage. Phosphorus promotes flower bud production and hardiness. Potassium is necessary for well being.

Weather. Cold weather can kill flower buds. Usually you see the brown buds in the spring. Cold spells in the fall or spring can damage buds that are not hardened off. Bud blast (blooming in fall or winter) uses up good buds which are then not available at the normal blooming time.

Age. Most rhododendrons take 2 to 3 years to bloom from a rooted cutting unless forced. Some take longer and some bloom sooner. From seeds the plant may take 1 or 2 additional years.

Exposure. Some rhododendrons need full sun to bloom and others can take fairly dense shade. In general, the more sun the more flower buds but also the greater exposure to damage from desiccation in summer or winter. More shade produces more foliage and less flowers.

Moving that many plants sounds like a major undertaking to me, and I'm not sure it would solve your problem.

From what you've described, your rhododendrons are not suffering from too much water - your plants look healthy but are not setting flower buds...Climates with high rainfall, coolish temps make them perfectly happy; they like moisture as long as the soil doesn't remain so soggy that root rot results, it's drought that can cause failure to bloom.

These are not among the plants that bloom better in response to stress. My best guess is still shade.