dead weight

Lack of recovery. You're probably trying to do too much or exercise too frequently - likely that your cardiovascular fitness is good enough for you to keep running past what your legs can handle.

Heavy legs during a training session is often a sign of lactic acid build up, but in this case (given you run every other day) I think it is more likely microtears in the muscles (normal muscle damage and repair).

My suggestion would be to find some alternatives to running if they are getting too sore. Perhaps some weight training, circuits, cycling, cross-trainer, or even something like boxing that is less hard on the legs - and alternating your running sessions with these other activities.