although the cog feels tight, it will probably be that it is not tight enough. If it was the cranks slipping then they will be ruined, it could possibly be the chainring if you have it the wrong way round and the chainring bolts are not falling flush into the recessed bit, that can cause movement when decelerating and accelerating. but i would guess its the cog, needs to be nipped up.
although the cog feels tight, it will probably be that it is not tight enough. If it was the cranks slipping then they will be ruined, it could possibly be the chainring if you have it the wrong way round and the chainring bolts are not falling flush into the recessed bit, that can cause movement when decelerating and accelerating. but i would guess its the cog, needs to be nipped up.