I'm pretty quick and usually can get on top of the net ... I do tend to rush sometimes when I sprint in because of my head bobbling and footwork.
Sounds like you are sprinting as fast as you can solely for the sake of getting close. You can't just sprint forward as fast as you can to the net and then pop into a ready position and react from there (sadly, I see this done a lot at my club).
Generally, yes, you do want to be as close to the net as possible, but more importantly you want to get as close to the net as possible
while under control and ready to make a play. That means you split step when the receiver makes contact, that means you are reading the returner's position and angles and know where you should be, that means you are in balance, your racket head is up, and your hands are in front of you ready to volley.
As a general guideline, if you are split stepping on/inside the service line and take that first volley a step or two inside you are in good shape. Obviously if you've got a big, heavy serve that produces a lot of short popped up replies you can go for the kill right away. Otherwise, the first volley is what sets up the point and you will probably see a lot of low volleys that you need to bend those knees and get down to dig them out or you'll be hitting a high mid-court volley, neither of which is that easy to put away. That's where you need to be comfortable having to hit 2-3 volleys to win that point and think about combos - e.g. get him to move sideways with one volley and then punch it into the opening with the next.
One more thing to add about coming forward. Remember you are on the offensive. Don't worry about defending against the lob, a lob won't hurt you. A true offensive lob (e.g. topspin lob dipping for a winner) is an advanced shot and tough to do on a return of serve. A high deep lob will get you off the net, but the points not over, you'll have time to go back and get it. It just resets the point to neutral and you can work your way back into control the point from there.