How can I avoid floating during short-field landings?
I did all my training in PA-28's (Warrior and Archer II mainly) and here is my advice) 

First off, practice makes perfect in all cases here. 

Second, I am not a CFI and you should ALWAYS seek the experience of a CFI to help you learn how to fly properly. Discuss everything with your CFI before trying anything. 

The PA-28 is one of the most benign airframes and even in the bumpiest the bumpiest of conditions you can fly it by the numbers every time. Floating on landing is generally caused by (in my experience) 2 things. 


Excessive speed on final - Remember you will only land nicely once you have gotten rid of all excess speed. This happens during your round-out and flare to landing. On a short field even a perfect landing will take up more runway if you are overspeed since you need to bleed that speed off. 
Ballooning - This is generally a result of over speed and over flaring. Since the PA-28 is a low wing (compared to the high wing of the C172) an over-flare to early will cause significant ballooning as you effectively bounce off the ground effect. A proper timed round out and flare will come in time. 


One thing you can do to mitigate this (and I found this helped as a student pilot for me) is to fly a longer final leg (and thus a longer downwind). When I was learning to fly a stable approach it took time for me to get the settings in and really nail the approach. A longer final will help you settle the plane into place as a student and really nail a maneuver like this. Generally tight patterns are taught but if its safer and better for you to fly a longer final and nail the maneuver, do it. 

In the PA-28's you can (and should) be right on 65 kts when doing a soft field. Practice descending at a constant airspeed at altitude with your instructor. 500ft/min right on 65Kts with full flaps, then come back into the pattern and use what you learned to come in right as you should. 

Cut the power a bit earlier. Generally you are most likely taught to cut the power over the numbers and touch down at the markers. In a short field situation you can cut the power a bit earlier, pitch down to keep you speed up, and touch down right on the numbers. You should have no issue getting the plane down in under 1000Ft.   

Train at an actual short field (or as short as you are allowed to fly). My flight school generally had us going between 5000ft airports with 2 runways (we were based at KPNE). Crosswinds were never really an issue and runway length left plenty of margin for error. My instructor made it a point to take me to 3000 X 60 fields to really get practice in (the shortest the fligh schools insurance would cover). Depending on where you are training out of this may not be an issue. 

One more general habit of new pilots is over correcting and getting your self into an increasingly corrective state. This happens both laterally and vertically but in your case the vertical over correction is going to be the killer. If you are comming in and notice that you are low you add power, pitch up a bit and now you are too high. You pull power and pitch back down but this builds up some speed. Now you find your self over correcting all over the place to get back to a stabilized aproach. This is where the long final will help. You need to really get the plane on that 65Kt mark, trim it off, leave the power alone and fly it in. If you get bumped by a gust or a thermal wait a second and see if the termal settles then and only then should you come in with a small correction.