Unfortunately this question is not a very easy one to answer because there are many options for heating and cooling your house.
In your question it seems like you have a working heating and cooling system so I would have to ask you if there was a specific reason that you wanted a single system tied together that both heats and cools your house?
In the warmer climates you often see a combined heating and cooling system because the central air conditioning equipment (indoor evaporator and outdoor condensing unit) can turn into a heat pump using a reversing valve. This allows the single unit to heat in the winter and cool in the summer. The reason this equipment is usually only used in warmer climates is that most heat pumps stop working effectively when it is below freezing outside (also the days that you need heating the most).
Because heat pumps don't work effectively in colder climates we are forced to use some kind of fuel burning system that will heat the home regardless of the outside temperature. In most homes this is done by using a boiler to heat hot water, then the hot water is distributed around the house to the heating equipment. The heating equipment could be lots of different options including a radiator in the room, radiant heating beneath the floor, or a fan and hot water coil unit.
In your question you say that you are feeling hot air coming out of wall outlets near the floor so I would guess that you have some sort of fan and coil unit in the basement of your home. It is possible to get an air conditioning unit with hot water coil attached to it so you can use the same duct work system to distribute hot air and cold air. I imagine that you would have to extend the hot water piping from whereever it is now to the new hot water coil by the air conditioning unit. Then you would probably have to change your entire AC unit (or at least the controls for the unit) so the AC unit could read the thermostat for the house and activate its fan and the new hot water coil that it is now connected to.
I personally like having the hot air supply come from the floor and cold air supply from the ceiling because it causes a better air mixture for the space as hot air rises and cold air falls. So unless you have a significant reason to change your current AC system I would caution you from reconfiguring your system as I have described here because I don't think that a common system offers many benefits other than a potentially lower initial construction cost. Since your system is already existing I don't see many benefits to changing it now.
If there are more factors driving this decision making process let me know and I will try to help you out more.