4:16a Therefore we do not lose heart.

Therefore, we (excl.) do not become dejected.
-or-
So we do not despair.

4:16b Though our outer self is wasting away,

But although our bodies decay,
-or-
Even though our bodies weaken ⌊with age and suffering⌋,

4:16c yet our inner self is being renewed day by day.

yet our inner being is being made new again each day.
-or-
yet every day ⌊God⌋ makes our spirits completely strong/healthy/good again.

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