Scientific evidence and decision making typically isn't done logically. Nearly everything, if not everything, is done as probabilistic analysis.

My question is, let's say I discover something new that is totally consistent with existing paradigms. Do I need to provide less experimental evidence than somebody who is claiming something that isn't?

I think your statement is more one of definitions of words and their meanings then once of science...

Lets say that somehow humans had not mixed baking soda and vinegar together before today. You decide to mix these things and get a mild foamy eruption that is cool. You do it twice and the same thing happens.

In this 'ordinary' example I would say you need less evidence to make your claims because the means of reproduction of the experiment and testing are within the reach of almost everybody. This could be quickly validated across the globe, and the reproducibility is evidence.

Now, if you require a billion dollar machine and a year of time to reproduce the experiment, you need to provide a lot of damned evidence so you don't waste a lot of scientists time and money that could be used for better things. Same with any experiments on people and/or the environment that can have effects that cannot be recovered from.