Please be more specific describing step by step what your issue is, because as generic as you are describing it, people perform this all the time successfully.
The formulation "can you load up a preset that you have made before, change it slightly and save it under a slightly different name?" includes an inaccuracy related to the two "it". When you load a Preset it loads all parameters of an effect stored in that Preset. And in this case of a Green Scream the list of parameters is the Effect Type=Green Scream plus the values of Drive, Tone, Mix, and Volume. As soon as this Preset is loaded, there is no reference left other than these parameter values. So you cannot change "it" and you cannot save "it". Loading a Preset is identical to selecting the Effect Type via Type Knob and dialing in those parameters manually. That's all. Loading it via Preset is just more convenient, if you require same parameter settings frequently. At any time when a Module or Section is in focus you can press Store, select to create a Preset instead of a complete Rig and then on the next screen edit the suggested name to anything you want and then complete the store.
Example: I open Module C, use the Browse Knob to select the preinstalled Preset "DIST Green Crunch". It loads Type=Green Scream, Drive=5.6, Tone=8.0, Mix=100%, Volume=0. I can now change any parameter(s) for example Tone=3.0 or leave all parameters as they are. I press Store, select "C Module", the edit page suggests the name of the Presets loaded the last time, in this case "DIST Green Crunch". Here I can edit that suggested name into "DIST Green Scream 2" or "Fitziputzli". The next Store will finalize the creation of a new Preset. A new Preset is born. If I now open Module B I can turn the Browse Knob and select "Fitziputzli" from the list of Effect Presets. "DIST Green Crunch" is also still available. "Fitziputzli" loads Type=Green Scream, Drive=5.6, Tone=3.0, Mix=100%, Volume=0, as expected. If I miss to edit the suggested name "DIST Green Scream", a second Preset will be created with this same name. I will afterwards find two Presets "DIST Green Scream" in that list of Effect Presets, one with Tone=8.0, the other with Tone=3.0. If you want to get rid of the original one, you need to use the Browse Knob to select that Preset, which loads Tone=8.0, and while you are on the Browse Screen press the Soft Button "Delete".