That shall soothe this restless feeling, (line 15)
Here's another definition of this feeling of sadness. It's not agony, or suffering. It's just a kind of "restless feeling," a sense that not everything is quite right. We're willing to bet you've felt this way, just kind of antsy and dissatisfied. This is another try by Longfellow to describe this very particular and kind of strange mood that the speaker is in.
The restless pulse of care, (line 34)
We like this image a lot. The rhythm of your pulse is supposed to be as slow and calm and steady as anything in your life. It's literally the beat by which we all live. So, when our pulse gets "restless," when it becomes jumpy and loses its rhythm, we know things aren't right. Again, this isn't pain; it's not the same thing as a heart attack. It's just the feeling that you could use some calming down, a little bit of soothing.