It's a definite adjustment coming over from a superzoom, but I think you're getting the hang of it. Depending on how you handle the DSLR, and how much you understand about the differences in the handling and processing, it can even seem initially disappointing. However, having spent some time now with my A300 and lots of shots under my belt with all of my lenses, I can tell you as one of the first H5 users in Dpreview, and one of the most avid fans and supporters of that camera through my 2 great years with it, that the A300 is better than the H5 in some areas, and can do other things the H5 or any prosumer simply cannot.
I think the 18-250 is going to help you overall - as Jerry mentioned above, it isn't going to solve your light sensitivity issues, but it will give you a more familiar range and flexibility that you were used to on the H5, with that big zoom.
Your issues with the shutter speeds are lens related, not camera related. The H5 has a maximum aperture of F2.8 at 36mm - whereas the kit lens is beginning a little wider but still only at F3.5 - which is two f-stops less sensitive. So the same lighting and same ISO would require a slower shutter speed to get the same metering - 1/500 on the H5 versus 1/250 on the A300, for example. As you use more telephoto, the H5 could zoom all the way to 420mm with a maximum aperture of F3.7. With the kit lens, you're at F5.6 even when you're at 105mm equivalent...if you were to zoom the H5 to roughly that same focal distance, you'd be in the range of F3.2 or so. Now you're a very significant 2 full stops of sensitivity different - a 1/500 on the H5 versus a 1/125 or even 1/60 on the A300.
But - that's where the ISO factor comes into play. You can get one of those stops back by increasing the ISO. The H5 remains usable at ISO400, but noticeably grainy...while the A300 can shoot in ISO400 if properly metered and be basically free of noise. At ISO800, the H5 was getting ugly - noisy, lots of smearing, and bad color fadeout...usable in a pinch, but not for everything. The A300 is still shooting great detail and perfect color at ISO800, only just introducing some minor noise. At ISO1600, still usable on the A300 - depending on your tolerance of noise and noise removal techniques...detail and color remain good.
As mentioned above also, the ability to swap lenses as your needs grow is a big advantage with a DSLR. As long as you're willing to spend the money and deal with the bulk, the DSLR's abilities are expandable with better lenses. Lay your hands on an F2.8 constant zoom, and you're at the same or better sensitivity as the H5 with significantly better optical quality and sensor size to preserve greater detail. Slap on a low light lens like an F1.4 or F1.7 prime, and you can shoot handheld snapshots at midnight at high ISO, or turn a grey post-dusk scene into bright-as-daylight and get all the fast shutter you need to freeze action when the H5 would be shooting shutters measured in seconds.
It just takes a bit of learning and adjusting...and choosing the right lens for your shooting. I think the 18-250 is a great all around lens that you should enjoy. Finding a cheap 50mm F1.7 is almost a rite-of-passage for Alpha buyers, and I heartily recommend tracking one down while they're cheap. The low light abilities of this lens combined with the high ISO is a blissful revelation.
As mentioned, avoid Auto mode like the plague if you can. P mode is nearly identical to A mode - so if you're still learning or uncomfortable with forgetting to make changes in A or S mode on the fly, use P mode. It will save your custom settings, and still let the camera choose WB, ISO, aperture, and shutter - but give you the ability to spin the control wheel to override the aperture/shutter combination to achieve a desired depth of field or faster shutter speed.
A and S mode are useful when you want to lock one of those parameters for a particular type of shooting, and M mode when you've got the confidence to set both for your needs. Even the scene modes are useful, as they allow you to change some of the preset parameters like focus mode, metering mode, WB mode, focus area, metering area, ISO, etc to fine tune it to your preferences.
Have fun, shoot alot, practice alot and get used to the sweet spots and handling of the A300 - and you should end up falling in love with it.
Hope that helps!
--
Justin
galleries:
http://www.pbase.com/zackiedawg 
