HC carpet + corys is not a good idea
HC carpet without CO2 is also not a good idea
Would remove the sword plant's for they will get too big for 10 gal and must have fairly deep substrate to do well.
Other than that, I would just plant the HC and continue what your doing.
Plant's don't care where they take nutrient's from and can do just as well with water column dosing as they can from enhanced substrate with exception of the sword plant's that really do seem to do better with both enhanced substrate and also water column fertilization.
Need three to five inches of substrate for sword plant's to get good growth potential which as mentioned,would soon result in them taking over the tank IME
+1 to everything roadmaster said
Your swords WILL take over the entire tank. Most high tech plants are not 'root feeders' like crypts and swords are. They will be fine in sand.
A few more things:
HC + corys is not a good idea. The corys will probably wreck the HC as it is a very delicate plant and difficult to keep anchored until established.
HC without CO2 is also not a good idea
Your substrate is not going to be the issue here. HC's roots really don't do much except anchor the plant down. Do not underestimate how good regular sand works. Unless you have something like ADA aquasoil then IMO sand is better than other substrates for various reasons. Easier to plant in, delicate plants (like HC) anchor more easily, etc.. Others may disagree but as someone who used to have eco complete I enjoy my blasting sand tank much more.
To have a chance with HC a good pressurized CO2 setup is almost required. Otherwise you will most likely not be successful. Substrate doesn't really matter (keep the sand)
Spend your money on a CO2 setup. It is far more valuable than higher lighting, substrates, etc