Having problems with reduced caloric intake... help?

are you eating back your exercise calories? or even half? or just at 1200ish calories a day, plus exercise?

also are you drinking enough water?

well if you're eating at a deficit on top of working out, and not eating back at least some of those calories, you may not be getting enough. even if you feel full, going extended periods of time eating that little plus burning up a decent amount of calories each day, might make you feel shaky and your body feel drained. just start eating a little more, and see if that helps with how you feel? just listen to your body and don't overwork yourself

That skakey feeling is your body telling you it's depleted. You need to eat more. Regardless of what the test told you your BMR was, it's obvious your body isn't in agreement. Eat back the calories you burn, and feed yourself more and the shakes should stop.

You're feeling weak because you're not eating enough, and because your BMR is likely wrong. I checked two online calculators, and both came in at 1465.

Calculate your TDEE, and try doing TDEE -10% and see where that leaves you.

You're not going to be able to lose weight aggressively because you're not that far from a healthy weight (depending on your frame size) so really your goal shouldn't be to lose more than 0.5-1lb per week.

Are you diabetic? I'd have my blood sugar tested.

If you weren't exercising before but you are now walking 3-5 miles a day, you likely could continue with the same caloric intake as before and still lose at a slow yet sustainable pace. You don't need to make so many drastic changes with relatively few pounds to lose.

You said you determined your BMR but maybe calculate your TDEE and take a 10% reduction from that since you prefer not to eat back your exercise calories.

Just to be clear - Bodpod did NOT give you your BMR.
It only gives a measurement of your bodyfat %.

From that, they then give an estimate of your BMR (or RMR, check the paperwork if they say Nelson RMR which several studies have shown is on low side of estimates) that is at least better than gender, age, weight, height based estimate, and should be within 5% of reality - if you haven't had any weight changes or tried to lose weight in last 6-9 months.

If indeed RMR - that is about 150-250 above BMR actually, use a site for getting Katch BMR using that bodyfat %.
If that was Katch BMR given, then bodyfat % is about 43%, right?

And if indeed much lower than other estimates - just means you have less muscle mass than average person your age, height, weight.
Which is probably not good at all - since it's the muscle that provides 15-20% of your resting metabolism.
So you really don't want unreasonable deficit that will cause more muscle mass loss - do you?

With that little left to lose - just use the tool correctly.
Be honest with non-exercise activity level. Really sedentary desk job/commute for 45 hrs weekly, and no kids, pets, many errands, ect?
Selected reasonable weight loss goal - 1 weekly until you get to 10 lbs left, then 1/2 lb weekly. You either pick it purposely, or for many the body forces that on then anyway.

Then log food accurately by weight eaten, liquids is for measuring. Calories is by weight, grams, not volume, cups and spoons.

Log your workout honestly.
Did you really walk 30 min at 4 mph the whole time, or really 25 min closer to 3 mph average the whole time?

Then follow the new eating goal that is given, which should be higher when you burn more that day in exercise, to keep the same amount of deficit each day.

Keep in mind you were eating 1800-2500 with NO exercise.
You merely must eat less than you burn in total, and now you ARE exercising.
So eating on exercise days 1500-1800 is still under where you should be.
500 less would be better now, so 1300-2000 on no exercise days - 1600-2300 on walking days.

You need to increase your exercise. Start some strength training -- free weights, NOT the machines, they are just a waste of time. With any luck this increase in exercise will help curb your desire to binge, since binging is often emotionally linked, and exercise = endorphin boost = happy brain. You may not think you are depressed or whatever, but let's be real here. It's pretty damn disheartening to be overweight, and I say that as someone with a lot of weight to lose as well. It may not be diagnosable major depressive disorder (and I say that as someone who actually HAS that diagnosis, I digress), but it still sucks, and it can definitely contribute to those emotional eating urges.

With your increase in exercise you can now, in fact, eat more! Way more, loads more. Feed your body, fuel your body. Focus on building muscle and being as healthy as you can be. It's a lot easier to keep going with your lifestyle change (or "diet") if you have other reasons besides weight pushing you forward. These can be things like living a long life, seeing your grandkids grow up, running your first 5k, finally being able to carry more groceries, or even knowing that if a zombie apocalypse were to break out, you'd have a decent shot of surviving thanks to getting in shape.

Basically, the more you exercise, the more you get to eat. And yeah, I know how daunting exercise seems. Any other time I have tried to lose weight in the past, I shrugged off the idea of exercising and stuck to focusing on my food intake instead, since we're all told that 80% of weight loss comes from your diet. Yes, this is true. But if you're someone who struggles with emotional eating, you need to exercise. Period.

I finally came to this realization about 5-6 weeks ago. I began working out again with the recommendation of my doctor as I have battled insomnia + MDD + PTSD for quite some time now, and exercise is good for alleviating symptoms of poor sleep and depression. Something just clicked I guess. I am now down 14lbs (in 6 weeks) and I am now STRUGGLING to eat enough calories each day. Yeah, that's right. I struggle to eat enough. I have been binge-free for 5 weeks now. It's a crazy turn of events for someone as overweight as I am.

I probably could've condensed this message, but rambling is what I do best.

Tl;dr: exercise more. ACTUAL exercise. You are 23 years old, unless you have had major surgery you have no excuse for not exercising. This comes from a 21 year old female who used to think that walking was appropriate exercise too after tearing my ACL two years ago. Nah-uh. You're young and you're not obese, so quit acting like it. And don't tell me you don't have time to go to the gym, because that's bs as well. I realize this all sounds kind of harsh, but I say it from a place of love. Do some cardio, lift those weights (heavy weight, fewer reps for females who want to be toned -- I can break down the science for you if you actually care), and then hey! You'll be able to eat more, and if you give it a few weeks, you may find yourself in the same position as me: down some lbs and struggling to eat enough.

Edit [1]: forgot to add that the shaking symptoms you are experiencing are from low blood sugar. This can be caused by either eating too many refined carbs (hellooooo sugar!), or not eating enough carbs (and by "enough" carbs I mean less than 35g of carbs a day).