I had similar problem with burning in my legs when I was first diagnosed with fibromyalgia 11 years ago. I distinctly remember my thigh muscles aching and burning as though I had been on a stairclimber or done squats for hours on end. It escalated from there to other body parts. I also had trouble walking and a feeling like my legs were filled with cement. But the burning was the first thing. Fibromyalgia is a rule out diagnosis, so you need to check other things but they do have a physical exam that they do that would lead to that diagnosis. If it is that, gentle exercise initially, like swimming or even walking through water can help tremendously. I was at my best when I got a really good exercise routine going, though it took me a while to get there. At first the drive to the gym took longer than my workout. I literally walked ONE LENGTH of the pool and had to stop. But I didn't give up and eventually had a 1.5 hour workout going 5 to 7 days a week. I hope you get to the bottom of your problem. Don't give up. Lots of things take a long time to get to the bottom of. And this is a great forum for support!!!
Judging from your description of your leg pain, I would bet that you are suffering from varicose veins. They proliferate during pregnancies, due to the weight of and pressure caused by the fetus. They don't go away! They continue to branch out and often begin to bulge, although sometimes well under the skin where they are not visible. They can cause excruciating pain when walking, climbing stairs, or during any other activity in which you are on your feet. Treatments are available. Some are better than others. Find a doctor whom you can trust. Don't sit back (or walk, for that matter), and suffer, without first making your best effort to improve the situation. Take action. Ask questions. Don't allow yourself to be ignored. Now, get going. Best wishes.