Not-good lighting conditions = need large lenses and imaging chips. Generally, "grainy" means poor lighting conditions.
Good stabilization = use a stabilizer (tripod, shoulder mount, etc) and never shoot handheld.
Shooting under shadow and bright backlight = use the camcorders controls (sometimes manual) for iris/aperture control. Most consumer camcorders have presets and at least one should deal with your lighting situation - each manufacturer may call it something different, but a "backlight" exposure preset is common is manual control is not desirable.
And... learn to use the white balance.
I *think* in your price range, the Canon HF S30 consumer camcorders have 58mm lens filter diameter and a nearly 1/3" imaging chip (single CMOS) specs. The high end of the Sony HDR-CX series will be similar, but over your budget. Compare these with your Panny's small 1/6" imaging chip and with no specification for lens filter diameter we can only guess (which I prefer not to do).
However, to be clear, even the 58mm lens filter diameter and 1/3" single imaging chip in the mentioned canons and Sonys will have issues because of the very compressed AVCHD format video captured by this crop of flash memory camcorders. Specifically, fast action and high compression (at the time of video capture) do not get along well. This has to do with the way the AVCHD compression technology works. The short version is that the "first" frame" is where most of the data resides - the following 7 frames are essentially "what is different" from the first frame. With lots of fast action, and keeping in mind these camcorders generally capture at NTSC standard 30 frames per second, 8 frames represents more than 1/4 second. A LOT can happen in that time. Assuming you go down this path, please be sure to capture at "best quality". This will result in reduced compression, better quality moving images and large file sizes.
There is a table in the camcorder's manual that shows how much space the various video quality settings will consume.
I regret that the manufacturers have decided for consumers what is "good enough". MiniDV tape's lower compression DV/HDV format is a much better fit for fast action. As far as I know, the remaining miniDV tape camcorder under $1,000 available is the Canon HV40. Its 43mm lens filter diameter is a little on the small (though likely larger than your current Panny) side and the 1/2.7" (nearly 1/3 inch) imaging chip is decent enough. Note that most prosumer and pro-grade camcorders continue to use miniDV tape, and while there are some AVCHD pro-grade camcorders, their "best quality" compression is a lot less than the consumer cams.
The HV40 and HF S30 have a viewfinder (and LCD panel) which many of the lesser consumer camcorders do not have (less $ manufacturing = less cost). From what I can get at sony.com, the least expensive consumer camcorder with both a LCD panel and viewfinder is the HDR-PJ760V which is well outside your stated budget.
Be aware that your computer may need some upgrade attention whether miniDV tape or AVCHD-compressed - assuming editing is desired...
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