Is it possible for the nose to become softer with age or from rhinoplasty?

You did not tell us how long ago you got your nose job, but for sure aging also affects the nose making it weaker and with droopy tip.

Generally speaking, many rhinoplasties involve removing segments of cartilage in an effort to make the nose smaller and more angular. By so doing, various elements of support are lost and your nose may not seem as structurally intact. Though the trend nowadays is to be more conservative and less ablative, this is not universally accepted. Additionally, there is some thinning of both the nasal cartilages and overlying skin as we age.

The short answer to your question, is yes...the nose will become softer with age.  It should not, however, 'lean to the side'!  You didn't say how long ago the surgery was, but I suggest seeing your doctor again or seeing another surgeon to evaluation.