I think the poor contacts and high capacitance between rows of contacts on the breadboard is causing your filthy, dirty signal.
I made all my prototypes using Veroboard stripboard where every connection was soldered and there were no wires all over the place. My layouts were neat, compact and tidy.
Even high frequency and high power circuits worked perfectly. The stripboard layouts looks good enough to be sold as the final product for many projects where only one custom circuit was purchased.