What to add to paper where another team just beat you to the novelty I want to know what the best option is for adjusting an article in prep under the following circumstances:

I've been working on this new model of a natural system for a while. I established the theoretical basis for it, and showed that it is in agreement with empirical evidence. I have a second draft at this point.

Then I find one of the folks I was already citing was also clearly thinking about the same exact things for awhile. Their paper is published, and they did an awesome job. Darn! But no hard feelings. 

However I still plan to publish my paper because
1) They did not provide the theoretical basis, so we have different justifications for the model
2) My model comparisons to the data point out different things (but together my paper and their paper both confirm the model)

What are the standard adjustments that can be made when this happens? Is it best to:
1) Devote a couple sentences to this
2) Devote an entire paragraph (or more?) comparing and contrasting our models and conclusions
3) Simply emphasize that my work is now an extension of their work?
It seems you have answered yourself: you found different contributions between your paper and the other one. It's up to you to decide if those differences constitute publishable material and are worth submitting or not really.

The best recipe on how to handle it is, as usual, honesty. You can simply say something like "our findings are in good agreement with recent research...", "recent research came to the same conclusion using a different model...", cite the other paper, compare both and justify why your contributions are important.

This approach is perfectly common in research. Not everything is about novelty.

The reviewers will give you feedback that will provide some clarity and help you how to proceed from there.