If you are on a tight budget, forget about "going full frame"... That will always cost more. FF cameras are more expensive, as are the lenses for FF. Besides, with all the choices you have for use on a quality crop camera such as the 60D, there isn't a lot of incentive to go FF, unless you plan to make really large prints.
For landscape, you probably will want a wider lens. The most affordable options for wider lenses on a crop camera are all zooms. Within your budget, I'd look for a used Tokina 12-24/4 or Sigma 10-20mm HSM (variable aperture version) or Canon EF-S 10-22mm. You should be able to pick up one of the Toki's or Siggies used for under $400, maybe as low as $300, leaving as much as possible toward your next purchase (see below). The Tokina is currently selling for $430 new, while the Sigma has increased in price to $600 (was under $500 just a few months ago). Among all the ultrawides, the Canon 10-22mm would be my first choice for image quality and overall performance, and has recently been on sale for $600, plus the cost of the lens hood ($33 for Canon, less for third party clones). Used ones should be a little less.... but due to the popularity of this lens it holds it's value well, so don't expect very big discounts on used.
Either increase your budget or live with what you've got for portraits for now and start saving up to add a Canon EF 85/1.8 when you can. New it's a sub-$400 lens (currently $340, plus around $25 for a lens hood that's sold separately). You also might be able to find it used for a little less. Again, due to popularity it holds value well and won't be discounted a great deal used.
The total for one of the wide zooms and the fast prime will end up a bit more than your current budget... $600 to $800 or a bit more, depending upon whether you buy used or new and how lucky you are finding good deals.
Some comments on to your list:
Tamron 17-50 non-vc - decent inexpensive walkaround/midrange, good image quality and f2.8 aperture, but not really wide enough for a landscape or long enough for your portrait needs, plus it has pretty slow focus.
Tokina 11-16 2.8 - sharp and well built, but prone to flare, higher priced and a very narrow range of focal lengths... all just to get f2.8 which most people don't need on an ultrawide lens anyway.
Canon 15-85 - a very nice walkaround/midrange zoom, wider than most yet with nice tele range and excellent image quality as well. But it's f3.5-5.6 (slowest at the tele end, where you will want large apertures the most)... it's good, but not really ideal for either landscapes or portraits... plus it duplicates other focal lengths you already have and is well over your budget.
Canon 85 1.8 - recommended for portaiture... see above.
Canon 70-200 f4 non-IS - this essentially replaces your 55-250, but without IS and with less range of focal lengths, at considerably higher cost. You might find a used one within your budget, but not new. Usable for portraiture, but not much more-so than your 55-250 and doesn't really address your landscape shooting needs. You mention sports/action, and might want to get this lens eventually for it's fast/accurate focus (though I'd recommend saving a little more and getting the IS version instead... it's a newer design and a little sharper than the non-IS version, too... rivals the latest and greatest f2.8 IS Mark II for sharpness, in fact).
Tamron 70-300 4.5-5.6 VC - same issues as the 70-200, except having VC is a plus, while the variable aperture is a minus. If it's a USD lens, it would be faster and more accurate focusing than the 55-250. If it's not USD, it will be slower focusing.
Canon 70-300 IS - same issues as the 70-200, except having IS is a plus, while the variable aperture is a minus. If it's a USM lens, it will have faster & more accurate AF than the 55-250. Though I know some folks who use 70-300 for sports in good light, unless you spend a lot to get the L version it won't be as well built or have the same AF performance as the Canon 70-200.
Canon 17-40L - Not very wide on a crop camera. I had a 17-35/2.8L that I sold and replaced when I went from full frame (film) to crop DSLR... it wasn't wide enough. By the time you are spending what the 17-40L costs for use on a crop camera, you really should get the EF-S 17-55/2.8 IS USM instead... it's a full stop faster, has IS, has a bit more reach at the long end, and has as good or better image quality. It's not as well built as the premium L-series lens, but is solid Canon mid-grade build. Either 17-40 or 17-55 are solild walkaround/midrange zooms... But they are only moderately wide on a crop camera, not really wide enough for landscape, nor really long enough to add anything to your portrait capabilities.
Canon 24-105L - Yet another walkaround/midrange zoom. Not wide at all on a cropper. And as an f4 lens, will leave you wanting a larger aperture for portraiture at times. Besides, it sells for more than double your stated budget. If you get serious, track down a good used 28-135 IS for around $250 instead... it gives you everything the 24-105 does for 1/4 the price, just has a variable f3.5-5.6 aperture and not as well built/sealed as the L-series (but not bad either, it's mid-grade build).
FYI: I am currently experimenting with a Tamron SP 60/2.0 macro/portrait lens instead of my 50/1.4 and 85/1.8. It's a crop only lens, but is the only macro with larger than f2.8 aperture, making for nicer background blurs in portraits (actually there is also the Zeiss 100/2 ZE, but it's manual focus only and very expensive). If you don't shoot any macro, it might not be your cup of tea. But for me, when I want to travel light it might replace three lenses in my camera bag... the two portrait primes and a macro. The only shortcoming I've found is that it's not fast focusing (it's not been upgraded with Tamron's USD focus yet), so isn't very useful for sports/action shooting. Cost me $400 after a $100 rebate recently.
In summary, I'd recommend:
For landscape choose from the Canon 10-22mm (best), Tokina 12-24/4 (2nd best) or Sigma 10-20 w/variable aperture (3rd best).
For portraiture, get the Canon 85/1.8. This lens has fast focus and also can be used for sports, when the focal length is enough.
Start saving up toward a 70-200 or other sports oriented lens, to upgrade from your 55-250.... but live with that in the meantime. Oh, and be aware you might have trouble getting the larger, off-white 70-200mm into concerts and sporting events (tho the f4 versions are a little smaller and less intrusive than the f2.8 versions). Some venues prohibit what they perceive to be larger "pro" lenses... While a more compact, black 70-300 or 55-250 might sneak by.
Eventually you may want a mid-range/walkaround zoom. Since you have and use several fast primes, you might not need an f2.8 mid-range zoom. You have several good ones on your list already, might want to add 24-70/2.8... Canon Mk II if you win the lottery... or Tamron with VC... the Sigma is popular and well-regarded, too.
Oh, and by the way, I frequently use tripods and monopods at outdoor sporting events. Not sure what your problem is with that.