I'll try to be as succinct as possible..
I moved into my house in 2005. Prior to this, the air handler had been replaced in 2004, and the condenser in 2002.

Since we moved in, during summer months the coils were freezing over every few weeks. I (probably irresponsibly) just took the cover off and pointed a hair dryer at it for 20 minutes and it'd work again. Well, this summer it was freezing over every day and I'd finally had enough. Called a service tech and they diagnosed the problem as having mismatched units. I have a 3-ton, 10 SEER, Trane condenser, and a 2.5-ton, 10 SEER, Goodman handler. As a stop-gap temporary fix, they put a bigger piston in the line, and now it doesn't freeze over anymore. But it doesn't get the house cool enough either, and it's always running - yay Florida heat! My power bill is insane.
Incidentally, the compressor is constantly sweating now.. Probably not a good thing?

So, I'm looking at remedying the situation by replacing the handler, but am being told that 10 SEER systems are no longer available. One AC company is telling me that you cannot match a 10 SEER with a 13 SEER, and another is telling me you can if you install a "Hard Start Kit."

I have no clue about this stuff - it's really not my wheelhouse, I'm just trying to absorb as much info as I can, research it, and make an informed & affordable decision. Any guidance any of you might have on any aspect of this issue would be appreciated.

The air handler is not actually SEER rated per say. Air handlers are rated by tonnage or BTU capacity. Actually all R-410A air handlers can be used with R-22 with no problem. The only thing you have to do is use a R-22 expansion valve to convert the air handler over.

Pistons or cap tubes do not offer the refrigerant flow control that a expansion valve offers and you will be surprised at how well a system works with a properly installed expansion valve. Now on some systems you might also have to add a simple hard start kit and time delay to complete the expansion valve change over. This simple system change has been done for years to help underperforming systems function much better.

As you mentioned your main problem is the fact that the coil in the inside unit is to small for the outside unit. Correct that by changing the coil or the air handler to the correct size or one step larger. You can always be a bit bigger but never smaller on indoor coil size.

Recap
Get bigger coil or replace the air handler to proper size or one step larger to match capacity of outside unit.
Add thermostatic expansion valve (TXV) designed for R-22 and properly sized for outside unit. (Yes you size the TXV for the outside unit not the indoor coil)
Add hard start kit
Add time delay to low voltage control circuit to prevent and quick restarts (not totally necessary but I use them)

But the biggest problem is yet to come. Find a service person who really know his stuff and is able to install all the goodies and charge the unit correctly.

When this is all done correctly your utility bills will be a lot better and the humidity level in your home will be lower and more comfortable.