Also don't forget that if you're doing landscapes, it's more than likely you'll be setting your camera up on a tripod in portrait orientation and will be doing some stitching. You can get away with this using your 35mm depending upon your distance from the subject.
Check this out bro $364 Excellent condition. I buy from them and know MANY others whom also buy from them. I in fact just bought an 75-150 E series manual focus Nikon from them the other day. : Tokina 17mm f3.5
Nikon Autofocus 17 F3.5 TOKINA ASPHERICAL ATX PRO (77) WITH HOOD, CAPS, 35MM SLR AUTO FOCUS SUPER WIDE ANGLE LENS - KEH.com
Granted, this is an FX lens so you might get a bit of corner vignetting, but I've seen images taken with an DX crop sensor and they're really nice.
As far as your initial choices go... you should also remember that the Sigma is not an f4 lens. it's 4-5.6 the other 10-20 they make is an f3.5 lens, but is also $650 vs.$479... I own an Sigma 17-70 and love its image quality, but it's a slow lens and the 2.8 it starts at disappears the second I move the zoom barrel. I want to sell it actually, in order to fund another manual focus lens.
So while it's cheap-ish, you're also losing a lot of light the moment you zoom in. And unless you're using a tripod properly, that will mean a LOT. Unless of course you're only shooting during the day and have decent light. This is part of the reason I say it's wise to save for good glass. If you do, you'll not ever regret it. Sure, it's more money initially, but less sour grapes later. That's worth the cash IMO.
On another note, if your primary concern is shooting landscapes then auto focus is a bit of a moot point. Most of the time you'll be shooting at infinity! Plus, you still have the camera's focus point indicator and your eyes... Unless you're blind and the indicator is somehow totally off, those two tools are sometimes more accurate than some AF systems.
That said:
Tokina 12-24mm f/4 AT-X 124 AF Pro DX Lens ATXAF124DXN B&H Photo
or of course the 11-16 Just sayin'.