Too high abv in a boiler is dangerous. And it can be the cause of the issues the OP is encountering. Not saying it is, but it may. 15 or 16 distillations should take you to 190 proof. What may happen, and i am not sure, but it can cause the issue of not reaching 190, is ethanol blockage. For a column to work well, it needs to be able to establish a temperature gradient. Higher at the bottom, lower at the top. If your still is charged with very high abv, it will create the lowest temperatures already low in the column, and this hampers the rising and further purification of the spirits. Since the boiler is still fed with power, a pressure situation can evolve where water i(or boiler contents) are pushed up the column, smearing into the more purified alcohol. To test it, dilute to 30% prior to doing the 15-plate run.
I think Southernhighlander's advice may work well, even though with a good system you wouldn't need that. Personally, I feel Liquid Management instead of Cooling Management works better. In LM you don't have a dephlag that is influences by water temp, water pressure, surrounding temp and air pressure. Instead you have a column cooler that liquifies all gasses and you then select manually via a needle-valve what goes back for redistillation and what is taken out as product. The advantage of this Liquid Management column management system for vodka making is that it is only influenced by differences in air pressure (well, the boiling point of alcohol is), so there is more control.
My experience with a lot of those that use packing in their column, is that it works nicely. Depending on what you use, it can give you like one distilation cycle per inch of column packing. Distillami, putting some scrubbers in won't do it. Look at something like our helicon column packing.
BCouts, I think the sizing question really relates to how much product you want to make. How much vodka. I know Liquid Management columns that are packed and that produce 190 at rates of 15 to 20 liter per hour. That's on just a 5 inch column diameter and 30% boiler charges.