Need Advise on Replacing Heat/Air Units

First off live in an older house...was built in the '60s....and I use oil for heat which was initially installed and am still using the same old clunker oil furnace. Evidentally when air conditioning was installed, NEW ductwork for wahtever reason was ran...and as a consequence for example it comes out from the ceilling, whereas heat comes out of baseboards and floor. Before last summer, I replaced the compressor outside and upgraded to a 5 ton unit....I had hoped that this would increase air flow but unfortunately minimal difference was noticed. The air handler itself is rather old as well although I have no idea how old. I just know its a 5 ton coil.
The main problems would have to be the air flow, not enough air coming through the vents during the summer for sure and the a/c just running constantly and in the winter, since oil heat is so hot, it will heat, just if more heat were coming out of vents obviously it would heat faster.
Questions are: If I replaced air handler, would this increase air flow in summertime and cool faster?
Should I clean out ductwork and possibly try to combine heat/air into one duct....?
And would upgrading the furnace to a new efficient system be the best option as opposed to running nat. gas lines and using that?
thanks for helping a young homeowner out as I am trying to learn this as I go!
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