It may also be helpful, if you are not already doing so, to bring satipatthana along with you as a daily practice. It is difficult to do this outside of a monastic environment, but one can begin by first anchoring mindfulness to routine tasks; brushing teeth, toiletries, bathing, and so forth.
This feeds into the seated practice, which feeds back into the day-to-day, and ultimately helps address Problem No. 1, in my experience. There may come a time when, instead of coming to awareness when it's already in full swing, you see it start
. It can be very encouraging to see such results directly for oneself.
As for the second problem, what I found helpful was incorporating the rest of the body; it occurred to me that there wasn't a breath, there was just breathing, and thinking of the whole body together in this way taught me that disengaging from the breath could happen in the same sort of way that one disengages from, say, arm control. There is a qualitative difference between moving the arm, and having it at rest, and it was a matter of 'doing' that with the breathing-body-part.
If that makes sense.