Well the short answer is start using alpha channels (no such thing as alpha layers, by the way). I realize from the way you described alphas that that's probably not what you wanted to hear, but bear with me for a second. Once you get used to making alphas, it takes 2 seconds to do, and you'll find it gives you ton more control over your image quality. You'll very quickly find that you no longer think of it as "having to get into all the gray scale work", but as a quick and simple step to insure that your work looks the way you want it to look. Also, if you ever want to branch out past SL, alphas are vital for many aspects of 3D and 2D work.
For an analogy, being a digital artist (whether professional, amateur, or hobbyist) and not using alphas is kind of like being a painter and not using masking tape. At first, all that taping might seem like a big waste of time to a new painter, but in the end, every painter discovers that it actually saves a ton of time, it improves the quality of the work tremendously, and it just becomes an instinctive part of the workflow. Trust me, if you're not using alphas, you're spending way too much time on certain tasks right now, and you're sacrificing so much control over your images, the amount is almost indescribable.
Anyway, as for your current problem, it sounds like there are two things going on, the white backgrounds and the black backgrounds. I'll start with the white, since that's the easy one.
It's perfectly normal for trapsparent areas to turn white after saving to TGA if no alpha channel is present. This will happen with all versions of Photoshop other than 7.0, including 7.0.1, and including 7.0 with the updated TGA plug-in. The reason this didn't happen to you in 7.0 before is because 7.0 was Adobe's one and only experiment with automated alpha channel creation for TGA. You don't actually see the alpha in 7.0, but it is there.
For 7.0, they tried to go for a more WYSIWYG feel (ironic, considering you can't see the alpha), but the experiment was a complete and total failure. The automation creates artifacts, the most common of which is a ghostly white halo around the opaque parts of your image. Under the 7.0 model, those artifacts are impossible to get rid of. Under every other version of Photoshop (and all other comparable graphics programs), those artifacts are easy to fix, and even easier to prevent in the first place. Just create an alpha channel properly, and the artifacts go away, instantly.
So, the way to get rid of those white backgrounds you're seeing is to create an alpha channel and make sure to include it in the TGA output by saving as 32-bit. If you want the instructions for how to do that, see
this thread. After the first couple times, it should take you no more than 5-10 seconds to fix each image, assuming you still have the original layered work. If you didn't save layered copies, then the repairs will take you a little longer than that, but for any new images, it should still only take a few seconds to add the alpha before saving.
Okay, problem number one solved. On to problem number 2, the black backgrounds. If I'm reading you right, it sounds like images that showed up as transparent on your old machine are showing up as black on the new one. That definitely sounds like a graphics card problem. It sounds like your machine just doesn't want to draw transparency, period.
I'm almost positive this problem is driver-related, and I'm sure it can be solved, but if it turns out it can't, return this machine, and get a different one, preferably a different model. This sounds like a pretty serious issue. Being able to draw transparency properly is a basic function, and I'd be willing to bet that if this machine can't do it, then there are probably other systemic problems with the machine that you haven't had time to discover yet. Don't panic yet though. It's very likely that this can be corrected.
EDIT: After looking at the system specs, I agree with Cottoneil. I'd be that the Intel graphics card is the culpret. They're really terrible. They're also not supported by SL, which means anything could happen when you run SL on this machine. Frankly, you're lucky it works at all.
As for the Radeon, correct me if I'm wrong, but the Radeon X 700 Pro an AGP card, right? According to the
spec sheet I'm looking at, your machine does not have an AGP slot. Unless I missed something, you can't install that card into that machine. You'll need a PCI Express card.