You can reduce ISO (usually reduces noise), close the aperture (increased DOF so more objects are in focus), or make shutter faster (can make motion choppy as you get beyond 2x frame rate); or some combination of the above. Beyond that you have to use ND filters of some type which allows you to open up the aperture and get shallower DOF if that effect is desired.

as far as video goes, how do you maintain shallow depth of field in the sun?

Put on a ND filter to block out light and open the aperture up. You can chose other ND's in increments, but 8 should get you a lot of latitude in direct sun. If you haven't used one of the DSLR's before, you'll be shocked how much DOF you will get even even with the aperture closed down significantly. Here's a quick video I shot yesterday in late afternoon without a ND filter and 7D on 720-60P with a Tamron 17-50mm mostly around f14 - 16.

Corey, back to your original question, both of those lenses are pretty cheap but from what I've seen both can be sharp when used in their mid ranges. I know one person using the Sigma and it works fine for him. I had that cheap Canon lens that came with a 20D a long time ago, and it was sharp at the wide angles but a bit soft when zoomed in to 50mm and focused in close. Beyond about 10-15 feet zoomed in and stopped down it was reasonably acceptable. But it has a really crappy manual focus ring and the best way to use it is on auto focus, even though it'll be slow to focus.

The Tamron 17-50 f2.8 is probably the sharpest of the cheaper lenses in that category. For your 28-300, if you have an LCD viewer on the camera, which you have to have for outdoor video, with practice you can probably do decent hand held shots from around 28mm to about 35-40 mm. Anything zoomed beyond that is going to get shaky without a tripod. Even with IS, you're not going to be able to do good handheld video at very long focal lengths. That's why cameramen use tripods.

Generally, you can live with the cheaper lenses for some things if you work within their mid ranges. In other words, don't shoot at wide open apertures, don't shoot zoomed in or out all the way, don't shoot stopped down all the way, etc. They can get acceptable images in the middle but usually have a variety of problems at the extremes.

Bill, I find the Tamron surprisingly sharp even at f2.8 and amazing as you get towards 5.6.