A Flight Instructor departing a non-towered airport reported they rotated early to avoid another aircraft taxiing across the runway.

Narrative

I was holding short behind Aircraft Y for RWY XX at ZZZ with my student. Aircraft Y took RWY XX and experienced an engine failure. Aircraft Y called 'ZZZ radio; I had another engine failure.' ZZZ radio then responded by letting everybody know there is a disabled aircraft on RWY XX. I back taxied on RWY XX to RWY XY while the Cessna followed us. Aircraft Y made another call stating he was going to wait for someone to come tow his plane at the end of RWY XX. My student then called ZZZ traffic announcing that we would be departing RWY XY at Taxiway 1. As we were rolling down the runway we saw Aircraft Y rolling down TWY 2. We never heard him make a call that he got his plane started; was cleared of RWY XX; was taxiing down 2 or that he was going to be crossing RWY XY. As we approached the intersection of RWY XY and TWY 2 Aircraft Y rolls out onto the runway; we rotated and climbed out early to avoid collision with Aircraft Y. The exact distance away from collision I am unsure about; I spoke with ZZZ Flight service and told them if we had stayed on the runway for 5 more seconds we most likely would have hit Aircraft Y. We rotated just before and flew directly over Aircraft Y on the runway I am guessing anywhere from 75-100 feet above him.