If the battery in my laptop is totally shot will that prevent it from powering up even if plugged in the charger?

It should have it's own internal battery. So if you plug it in, it should work. You can't recharge the battery, but you should be able to use it when it's plugged in, until you buy a new battery.

If it doesn't work plugged in, it's not the battery, it's a bigger issue.

If it is still plugged in with the power lead , it should work even though the battery is stuffed unless there is a lock on it.