My tender age in sorrow did begin (11)
Gosh, Adam really ruined it for the rest of us—at least for Christians, like the speaker, who believe in original sin. This idea assumes that Adam's sin is passed down like a chromosome. Everyone inherits it at birth, meaning that everyone is born with at least some sin and enters a world where the sorrow only increases from day one.
And still with sicknesses and shameThou didst so punish sin (12-13)
Herbert doesn't make it clear whether this "sin" is original (inherited from Adam) or the speaker's own doing. But to God sin is sin and must be punished. Sickness is part of living in a world that is Not the Garden of Eden (we have microbes now), so that's Adam's fault. And the speaker's shame could apply to either the universal badness of life (thanks Adam) or to his own wrongdoing.
With theeLet me combine (16-17)
These may look like throwaway lines, but they're actually theologically essential to Easter and this poem. On his own, the speaker is thin and sinful. It's only when he acknowledges this and asks for God's help that he can be saved.