From a PM:
... I noticed that this is a different style of tremolo than I have seen on Valve Wizard website. It takes up both halves of a triode and my understanding on ones ive seen is they do one. ...
I have since put most values back per schematic. Even if this is a simple fix im curious if I should try the tremolo circuit shown below? It uses less components and seems less complicated? Just for the sake of fitting it in my chassis better than the original.

Jack, Terminalgs and Sluckey already gave you the answer, but I'll summarize them again I one post.
If you carefully draw the Valve Wizard trem and your own trem on a sheet of paper, you'll see they are identical circuits, except some minor parts value changes which will slow down your trem (probably a good thing). Plus you have that 2nd triode.
The 2nd triode has its cathode connected to the cathode of the 1st gain stage; this is the method and location of the trem being injected to effect the guitar signal. The upside of this method is it's easy and checp to incorporate. The downside is that
strong trem depth will always cause background pulsing or pumping. It's normal and only avoidable by turning down the trem depth.
If you look again at the trem circuit you provided from Valve Wizard, it's just the oscillator, showing a plate output connected to nothing. In other words, it's not showing how that oscillator signal is couple to the guitar signal, and isn't usable by itself. You would still need a place and a method to inject the trem.
For grounding: on paper, the Gretsch's trem circuit is way away from the preamp, but that's just drawing for clarity/readability. If you follow the plates of the trem circuit back to the filter cap feeding them, you will find it's the 10uF cap furthest from the rectifier. This is also the same filter cap feeding the 1st preamp tube. The 1st preamp tube, trem tube and that last filter cap should all have their grounds directly connected together.
As for the noise: it's unavoidable in this amp circuit, and with this trem injection method. That's why this is an inexpensive student amp (among other reasons, like low power, few features, likely original small speaker, etc). To cancel out most of the trem noise, you'd need to switch to push-pull output tubes and inject the trem at the phase inverter (as in Fender's patented tremolo circuit from the 5E9 Tremolux), or by modulating the bias supply as in later Fender amps like the blackface Princeton. Those inject a common-mode trem signal into a push-pull circuit, so that the trem effects the signal, but the common-mode background noise is rejected.
The downside to the above is you're making a louder amp to start with, and yet you might find you're limited to 6V6's (with very deep trem) or 6L6's (but probably a weaker trem depth).
All this said, if you have a noise that sounds like a distinct "Put-Put-Put-Put" without being a pulsating background noise sound (or the difference between a ramp waveform vs. sine wave), then you probably have poor ground connections especially in the power supply and filter caps (though insufficient power supply dropping resistor size could also cause it). I'm describing motorboating, which you could diagnose by pulling the trem tube from the socket. If the Put-Put-Put is still there, it can't possibly be due to your trem, and is low frequency oscillation through the power supply.