I'm not a doctor, but I think what is happening is this.
At your early stage, your body is still producing some dopamine. As the disease progresses the short-fall increases and it is common for the medication to be increased over time.
Rasagiline is a long lasting drug with a fairly constant benefit. 1mg of rasagiline is "worth" about 100mg of levodopa in Sinemet.
I don't think the rasagiline is causing your stiffness. Rather it is likely to be reducing your stiffness, but unfortunately not by enough to eliminate it.
You report you are OK between 7 and 10 in the morning. This I suspect comes from your body storing dopamine while you are sleeping.
The Sinemet is likely to reduce the stiffness, but it is short lasting, so you may have "off" periods between doses.