Foot slappers usually have weak tibialis anterior muscles.
This muscle is the one to the front and slightly outside of your shin bone. It's the one that bulges up if you put your foot flat on the floor and then raise the ball of your foot whilst keeping your heel on the ground.
The classic way to strengthen this muscle is to sit on a table with your feet dangling and hook the handle of a medium size tin of paint over your toes and do some raises. Don't overdo it though because if you run with tired tibialis muscles you can get injured.
As well as controlling the front of your foot during the landing phase of running, the muscle has one other function - to lift your foot clear of the ground after it pushes off. If you take small strides you will be using the muscle much less. This may be why it's weak. Perhaps your stride has become shorter over the last couple of years? You might want to do some general leg stretches to lengthen your stride a bit.
I don't really get involved in discussions about the pros and cons of heel vs forefoot striking. It's all a bit unscientific.
In my view providing you do at least one or two runs a week outside you can pretty much treat treadmill running equal to outdoor running (except you'll have to use your imagination to visualise that car pulling out on you from a side street and the rabid dog from the estate chasing you).
There is a slight difference in action between the two but it is so small that it hardly matters, and this can be ironed out by doing at least one run a week outside.
People really dis' the treadmill but it can be your friend [sic]. You could even try running backwards to strengthen the opposing muscles. No don't do that; cos then you'll be facing all the other muppets.