Kenmore ... definitely change the needle, yes, an old needle can cause the thread to break. There could be a burr on the eye.
When I chain piece I always finish with the needle down. As soon as one piece is finished, I lift the presser foot slightly (enough to slide the fabric under, but not enough to disengage the tensioner), and slide the next piece until it touches the needle making sure to also slide it far enough to the right to get the correct seam allowance. I usually have just enough space between pieces to snip the threads.