Yes, you probably hardened it but how much is questionable.
Different hardenable steels have different needs when it comes to quenching them from a bright red heat.
The high carbon steels respond best to a water quench.
Many of the alloy steels will break if water quenched and need a oil quench.
A few rather exotic tool steels will harden just by cooling them off in air.
Some alloy steels are also quite fussy about the temperature they are at before they are quenched.
Chances are pretty good that your Dixie steel is a simple high carbon steel. If so, water would be the best thing to harden it fully.
This is not to say that an oil quench won't harden it some. Perhaps just right for a frizzen but chances are pretty good that it was hardened to a lower hardness (Rockwell) than a frizzen should be.
The baking of the steel after hardening is called tempering. It's purpose is to soften the hard steel a bit so it is not brittle.
Your 375 degree F bake should toughen it enough to remove its brittle nature but it was probably not needed.
When the steel got hot enough to silver solder (I am assuming this is a low temperature silver bearing solder and not the silver braze that needs a red heat to melt it?), it was hotter than your ovens temperature so it became the "tempering temperature".
Tell you what.
Before worrying too much about the hardness, do a poor mans hardeness test on it.
Using a good fine tooth flat file, try to file the edge of the new material.
If the file just skidds and leaves a bright area it is probably good enough to make sparks for a while with out wearing out.
If the file readily cuts into it, it is too soft.
Even if it is soft, it will produce sparks. It will also be easily gouged by the sharp flint and wear out rapidly.
Being as hard as it can be, the steel won't make any more sparks but it will not become gouged and it will last forever.
The trick is to get it hard enough to resist the gouging from the flint without being so brittle that it breaks.
