Should I be losing weight or building muscle?

With that much exercise, even if the rest of your day is sedentary your TDEE should be close to 2,000. If you're gaining weight on 1,200 then you're most likely making big mistakes in your tracking, whether from incorrect entries, incorrect portion sizes, not counting alcohol, condiments, forgetting cheat meals/days, etc.

I'm guessing you're fairly lean so unless you have noob gains to come from lifting, you probably aren't going to build much if any muscle while going from 120 to 110.

Your first priority should probably be figuring out why your tracking estimates are so off, and halt the ongoing weight gain.

Given that it's only been 2 weeks, I don't agree with the above, even though I often agree with Tom.

Since it's only been 2 weeks, I don't think you've truly tested your new routine enough to know the average effect. (What he says may apply if you don't see loss after 4-6 weeks-ish.)

A couple of things:

You significantly increased exercise. That's typically going to add water weight (for muscle repair). It can be several pounds, and that can mask gradual fat loss on the scale for a surprisingly long time.

You're a 22 y/o woman. I'm betting that means you have monthly cycles. It isn't the statistically most common pattern, but there are women here who've said they only see a new low weight once per month, at a certain point in their cycle. Most everyone who has cycles sees strange water fluctuations from hormonal effects, though the exact pattern differs from one woman to the next. You'll learn your personal patterns after you track for a few monthly cycles. For now, looking over just two weeks, there could be a hormonal water weight distortion in the picture.

Water weight fluctuations tend to be relatively big and quick, while fat loss is gradual and a smaller change daily. It's fat loss we really care about, right? Plus water weight fluctuations are part of how a healthy body stays healthy, so we don't want to try to defeat or game that.

In case you haven't seen it, this is a good read (especially the article linked in the first post), highly recommended:

https://community.myfitnesspal.com/en/discussion/10683010/the-weird-and-highly-annoying-world-of-scale-fluctuations/p1

I know it's frustrating to work as hard as you are and not see progress on the scale, but IMO your best strategy right now is patience for long enough to see a reliable trend.

Ideally, go for long enough that you can compare your body weight at the same relative point in at least two different monthly cycles. Since you ramped up exercise quite a lot, you might want to go even a little longer.

P.S. I get why you might reduce fats, because they're calorie dense. But don't go toooo low. Fats (certain essential fatty acids) are essential nutrients. You need to eat a certain minimum of those for cell health, absorption of fat-soluble vitamins, smooth digestive throughput, and more. Carbs are the macronutrient that are more flexible, can be fine-tuned based on your personal energy level and appetite reactions to them. (They aren't an "essential nutrient" in the technical sense that protein and fats are.)

It's calories, not macros, that are the direct factor in weight loss. Macros can have an indirect effect on weight through satiation or energy level, and are important for nutrition/health, but calories are the fat gain/loss determiner.

Best wishes for success!