As described in your models, if x3 is a relevant variable Model II will have both a higher R Square and higher Adjusted R Square than Model I. Also, Model II will have a lower Standard Error than Model I. Thus, you should keep this x3 variable and chose Model II.
If x3 is not a relevant variable Model II will have a higher R Square, but a lower Adjusted R Square than Model I. It also most probably will have a higher Standard Error than Model I. In this case, you should exclude the x3 variable and stick with Model I.
That's kind of the basics. In reality, once you add a few variables the added explanatory power of adding additional variables increasingly diminishes. That's even though those variables are deemed relevant and that your Adjusted R Square keeps on rising. However, let's say that adding x3 would cause your Adjusted R Square to increase by 0.15; that's a lot, and you would definitely keep x3. Now, you add another variable x4. And, the resulting Adjusted R Square increases by only 0.03. I think many people would not add x4. It may not be that worth it. Adding it may lead to a model that is overfit. You can test whether a model is overfit by using a Hold Out sample. The latter is probably more important than the ultimate level of your Adjusted R Square.
Going back to your two models (I and II). You should actually test both of them to check their performance in a Hold Out sample. Only after doing so, can you be sure that Model II is better instead of simply being overfit (which the Adjusted R Square will not capture that).