Less overall exposure and shoot at a more moderate aperture, say f/8 or f/11. try a little fill-in flash with the built-in, but run it second-curtain, -1.5 to -2.0 AND add a Full CTO filter (or any orange colored filter you may have). Bracket a lot to keep color in the flame. Try a lower ISO such as ISO 200.
Here's an exercise for you.
Set the camera at ISO 400. Daylight color balance. Jpeg files (this is a test at this time), Manual exposure. f/9.5 (somewhere between f/8 and f/11). Shoot a series of photos at these shutter speeds: 1/125, 1/60, 1/30, 1/15, 1/8, 1/4, 1/2, 1", 2", 4". This should give you plenty to look at and see what is happening.
Try the same, this time for "Tungsten" balance. Everything the same.
Do it again with the flash, say second-curtain and -1.5.
Now see which one looks the best. If they are all too light, lower the ISO from say 400 to 100 or stop-down the lens from to f/18 and pick the shutter speed that was the darkest. If they are too dark, raise the ISO to 800 or 1600 or open the lens to f/5.6 and choose the lightest image. Somehow, I think the first set of exposures will get you very close!
Take your best image and see what the settings are. Use them to do another test to "fine-tune" your exposure. Try these on "RAW" so you can fine-tune the images and also see how much more you can pull out of the file compared to the jpeg image. This time, bracket in 1/3rd or 1/2 stop steps. Try varying the flash-- -1/2, -1, -1.5, -2, -2.5, -3. Try it without the filter over the flash.
This exercise should give you a lot of info on how your camera records and get you better skills to both capture a great image and fine-tune it more to what you saw!