Last year I bought a cheap little azalea.  Then in July it dropped it's leaves, possibly because I let the soil dry out, now over the past week it's started to bud lots of new small leaves, does this sound like normal behaviour?

Letting to dry out meant that the tree dropped leaves, but you did not quite kill it, and it is starting to recover. No it is not normal for it to get dry in summer. Keep it watered and it should beok.

Hmm, It droped its leafs in june? Now it is september. That is 2 months or more without leafs!! If you are in the nortern hemisphere we are aproaching winter and your tree is wasting all its energy on pushing new leafs when they should really spend it on preparing for dormancy.

It is not often I dissagree with Jeff, but this is the one time. I do not think your tree will be OK.

No problem, but I have had an azalea that lost all leaves due to lack of water, it took a long time but it did come back. No it may not survive but it has two chances.