Just started working out, should I increase my calories?

Sure, eat more calories. At your current size/age, we'd expect your sedentary TDEE to be around 2300 or so, and could be high 2000s or even 3000+ with the exercise. Slower but sustainable loss is IMO better than feeling ravenous all the time. If you keep your deficit about the same size, your loss rate should be about the same, i.e., if doing lots more (in calorie terms), eat more, keep the same deficit.

Example: If you've been losing 2 pounds a week without exercise, that implies about a 1000 calorie deficit; if eating around 1300, that's implying a TDEE around 2300. If you added 500 calories of exercise daily (picking a number out of the air just to be an example), that would increase TDEE on exercise days to 2800, so eating 1800 would keep the same 1000 calorie deficit, implying 2 pounds a week loss still. (You'd probably make better fitness progress, and possibly also better muscle/strength gain progress, with a smaller deficit, besides. Muscle mass gain in a big calorie deficit is going to be limited, realistically, but strength gain is very viable, and some mass gain may be while still having body fat energy to draw on, assuming your protein intake and programming support that.)

If your strength workouts are similar on successive days, I'm also wondering if you have enough muscle-group-specific recovery built into your schedule for best fitness progress. On top of that, if you went from pretty negligible exercise to the current level all at once, that's pretty abrupt: Usually, a gradual ramp-up is a better idea for both fitness progress and impact on recovery, appetite, etc.

A big increase in exercise is an increased physical stress, on top of a reasonably aggressive calorie deficit (already a physical stress) . . . high cumulative stress can potentially spike appetite, and some people do find that particular exercise modalities spike appetite more than others (without necessarily being stress-triggered).

In the first few weeks of greatly increased workouts (especially muscle-challenging ones), you may still be seeing some water retention on the scale, masking part of increased fat loss. What I mean by that is that if your loss rate hasn't increased with the increased exercise, that may not mean there's not increased fat loss. It's just hard to tell, because the water retention (for muscle repair, basically) can muddy the results for a surprisingly long time.

Just my opinions, though, based on experience and reading. (I'm F, 66, 5'5" so close to your height, active, formerly obese. 1300-1500 gross calorie intake for me would be brutal, frankly, and would've been during weight loss: I started a bit lighter than you are currently, but lost well on similar (or a bit higher) calories plus all carefully-estimated exercise calories.)

Definitely eat more. Feeling like you're starving makes it much harder to stick with your long term plan, plus it may lead to not having enough energy to work out. As Ann noted above, doing HIIT training every day doesn't allow your muscles to heal between workouts. That could lead to injury. You would do better to mix up your exercise more so that you use different muscles on different days.