I am in the same boat, except my budget is about $300-400
From my research, it seems the 55-250mm STM have improved image quality than the 55-250mm IS II and is pretty much on par (some say better) with the canon 70-300 IS USM but price-wise these two are similar. The 70-300 IS USM is a much older lens but it is FF compatible whereas the 55-250mm STM is not. Build quality-wise the 70-300mm is much better (metal connector vs plastic connector).
Tamrom 70-300 VC version apparently is better than the other zoom lens you listed and is FF compatible. If you have the budget go for this.
I am personally still undecided, since the 55-250mm IS II can be bought for fairly cheap brand new and I dont know if I will use zoom lens that much. Perhaps I will hunt around for a second hand canon 70-300L or 70-200mm F4L which fall in the price range, but then again, wishful thinking.