Okay I own or have owned the 55-250 EF-S (not relevant since you want full frame) the Tamron 70-300 VC, the 70-300 L and the 100-400 L. I have also rented the 300 L f4 and the 400 L f5.6.
The 55-250 EF-S, the 100-400 L and the Tamron 70-300 VC are all very good lenses and generally quite sharp.
The 70-300 L is sharper, weather sealed and generally a better lens overall, but we are talking margins here. Is it worth the price? Probably not. But I wanted it badly and bought it anyway. I don't regret it. I use it along with my 15-85 as a two-lens kit that covers almost every situation.
The 300 L was sharp, nice and a stop faster so it took a 1.4x converter. But, it was shorter than I wanted and not as flexible without the zoom.
The 400 5.6 L is light and sharp but doesn't have IS and since it isn't a zoom it also isn't as flexible and it takes up a lot of space due to its length.
While the 100-400 L is not quite as sharp as the 70-300 L, it is sharp enough and the extra 100mm is pretty critical for shooting critters.
The 70-300 Tamron is as sharp as the 100-400 and the 55-250, which means it is sharp. It does have a tendency to hunt a little on autofocus sometimes. Not sure what the problem was and it may have just been an anomaly. A minor nuisance, not a deal breaker.
Since the 55-250 doesn't work for your needs (full frame) I would say the lowest cost solution is the Tamron VC. Is it as good as the Canon L? Obviously not, but it's almost $1,000 cheaper. If you don't need a zoom, then maybe the 300 f4 or the 400 f5.6 would work for you, but for the marginal difference in price, I went with the zooms.
I got my 100-400 L as a refurbished. They haven't had it in stock for awhile, but there have been some good prices lately, still, it is also about $1,000 more than the Tamron.
All in all, I'd say the Tamron is the best value 70-300 lens out there next to the 55-250 (which you can't use.) If you need longer length, you are going to pay for it.
I thought the opinion of someone who actually has owned or rented these lenses might help.