I have measured 122 Volts at .68 amps at the wires coming
into the thermostats.

You are misinterpreting this. What you just measured is that your
existing thermostats draw 0.68 amps of current. Your house electricity
is right about 120V, as it should be, and this is irrelevant for your
situation.

If you put your new 1-amp rated thermostat on, it will draw whatever
current it needs to function. The 1-amp rating is telling you that it
will pull something around 1-amp or less. Probably right around 0.68
amps would be my guess, actually. It is a "minimum power rating",
because it is telling you that your wires had better be able to handle
a minimum of 1-amp, otherwise it might be unsafe. You can't measure
current capacity in the way that you are doing. What you do instead is
look at the wiring and make sure it is large enough (and installed
properly) to handle up to 1-amp.

[ I'd be real surprised to find any 120V wiring in a house that is
rated anywhere near as low as 1-amp, but don't take my word on that. ]