Need Help with Weight and Stamina

Have you taken a good look at what you are eating?
Make a food diary and track everything that goes into your mouth for a week.

I have begun a low carb keto diet - and I feel pretty good.
My doctor is very happy with this.
I don't eat sugar or any sweeteners (even artificial), no starch (potatoes, corn, etc), no alcohol, no pasta, no bread, very little fruit, no juice, no soda - nothing that turns to sugar in the blood.
While it's hard to think of what you CAN eat at first - there are plenty of options.
Meat, fish, eggs, cheese, dairy products, avocados, olives, low carb veggies (spinach, asparagus, broccoli, cauliflower, etc), leafy green salads (watch the carb count on some dressings), etc.

I've started dropping some weight - which is something I haven't been able to do because even if I've counted every single calorie it seems even one piece of bread keeps my body from burning any fat.
When you don't eat carbs - your insulin doesn't rise and fall - the cravings are gone and when the cravings are gone - it's easy to stick to it.
It's been 2 months so far and I think this is a good change for me.
It's something I can live with long term.

Additional:
Hypothyroid will definitely make you tired and the only way to deal with that is to get it treated.
You will have to get your thyroid levels tested and then you need to take a thyroid medication once daily for the rest of your life.
While mild hypothyroid is merely annoying - if it's got worse and not being treated, your health could suffer greatly.
Maybe your local health department can help you with affordable doctor visits/tests.

When my kids were very little I did workout videos at home. I have fond memories of jumping around in the family room while they crawled and played around me. I had some 20 and 30 minute aerobic type videos, including a kickboxing one that I loved, as well as some shorter ab, arm and leg ones. You can find tons of these online (Denise Austin is a good one.)
If you start doing something like that you'll start feeling better pretty quick.
As far as food goes, cut those carbs! I know, I know, I love them too but nothing makes a bigger diet, weight and energy difference for me. Also an online calorie counter is helpful. I did that for a while and was shocked at how many calories I was consuming every day even when I thought I was eating fairly well. When you realize you'd have to run for an hour to burn off the calories in a Snickers bar it makes it easier to say "no thanks" to the Snickers bar :-)

Have you talked to your doctor? I sometimes get anemic. It triggers a vicious cycle. When I get tired from the anemia, I crave sugar and carbs for the energy because I'm tired. Then because I'm eating too many carbs and too much sugar, my blood sugar spikes and crashes, making me even more tired. Plus that means I'm not eating well, which makes the anemia worse, then I'm more tired, then I crave sugar even more...

I can interrupt this vicious cycle by taking an over the counter iron supplement. In just 2-3 days, I feel like a new woman. But, iron is dangerous if you take too much, so you don't want to just jump in and start taking it. Go to your doctor. Get a blood test. Have them check for anemia and for low vitamin D, plus your thyroid, and other common causes of fatigue.

A constantly run down feeling, even after a good night of sleep, is NOT normal. Your body is trying to tell you something. Please listen to it.

I feel like you described my life. I finally figured out two things: walk/move/exercise when I can AND eliminate carbs. It totally helped.
Especially as we age, limiting carbs is essential.
So.....I try to eat 1 main carb (things we think of as carbs: bread/tortillas).
I will eat a deconstructed sandwich for lunch: cheese & sliced turkey.
I eat more salads and make sure it has a protein: chicken or turkey
If I take the kids to eat a burger, I ask for mine to be wrapped in lettuce.
I do a lot of exercise videos at home because I can do them when the kids nap.
I try to take advantage of their down times: school, nap times for exercise AND
rest.
If the oldest child has a baseball game, I walk around the field while they practice after I talk to the moms for a bit.
I have a treadmill in my house so I use it as often as I can. I go for 30 min
intervals but have had to do just 15 mins due to time constraints (youngest
wakes up from a nap).
I try to do sit ups, pushups and crunches every day.
Do light weights in the morning when you watch the news: 5 lb weights.
I bought a few good workout dvds from Target so when I have chance, I do
those.....it has really helped.
Sometimes when hubby gets home, I try to get out for a quick 15-20 min
fast walk around the block.
The point is....everything helps. Do what you can, when you can. Steal moments here and there. It all adds up.

I'm thinking if you upped your protein maybe - so with your carbs (I like the idea of limiting them as B says, as too many is not a great thing), have some peanut butter with crackers, or have cheese and crackers, like make sure you're getting a healthy snack - with some protein in the morning between your meal. Fuel up.

When I did that, changed how I thought about food - as fuel, it helped. So instead of carbs (which we reach for when our sugars run low), protein keeps our sugars level (I think). I am not an expert and there are moms on here who know way more. So I'll let them answer. I just know what worked for me.

So a handful of almonds - kind of thing. Maybe keep a bag on you for your walk with your son. That will keep you going.

So if you add in some protein with what you are already doing, you'll feel some additional energy.

Sleep is huge. I think you said before your son isn't sleeping? So if you're not getting rest, you'll feel drained. Make sure you rest when he rests, and don't run around doing chores.

You might have to up the cardio a little bit. I bought one of those little home machines - like an elliptical (cheap one) just to do 15-20 minutes every other day. It helps. You can get them super cheap, small (apartment size ones), that are quiet.

Breastfeeding makes you ravenous - if I recall - so just recognizing that, and being ok until you're done - it will likely come off later. I was famished all the time. It did come off for me.

Good luck :)