Are you taking alphas whilst buffed? I think you can play to two styles; one where you are the Brute who shoots off ahead and spends some time in a group before the team show up, as soon as they show up shoot off again and leave them to deal with that group. (With a Dom the npcs shouldn't follow).
Or play the Brute who builds fury up with the lighter quick recharging attacks and then from moving from target to target maintains it throughout the mission.
One version is a bit like playing a Tank and the other one is a bit like playing a Scrapper.
Fury wears off too fast imo. A Brutes fury shouldn't wear off fast.
Edit: Almost constant use of Brawl is good and doesn't even spoil a Dual Blade combo.
He will honor his words; he will definitely carry out his actions. What he promises he will fulfill. He does not care about his bodily self, putting his life and death aside to come forward for another's troubled besiegement. He does not boast about his ability, or shamelessly extol his own virtues. - Sima Qian.
There is another thing that can help...Taunt.
That's right, I said Taunt.
Keeping aggro helps Fury fueled. In large groups, aggro can easily get split up. Firing Taunt can help you keep bad guys beating on you, thus keeping your Fury up.
Before Dual Builds, Fury was a debatable pick. Now, it really isn't. I have 2 builds now on all of my characters, the Solo Build and the Team Build.
Consider getting Taunt.
Quote:
| on the times I do fill it (the Fury bar) up (which are rare) by the time the lot of people move to the next spawn it's almost completely gone. |
Looks like you team up with slow people. Possibly turtles, snails or bAEbies. Solution?
Team up with faster people.
Hit things as often as often as you can and hit as many as you can and when there's nothing left to hit move on your team will soon catch on and start following your lead especially if your making things dead quick,
Moving from spawn to spawn shouldn't be depleting your fury. I usually go to the next spawn when there are only a couple enemies left in the first one. If you have trouble taking alphas, I suggest you stock up on some purple or orange inspirations. I find on my shield brute if I pop one or two purples for the first couple mobs the team usually catches on and keeps up with me the rest of the way so I don't even need to pop those insps anymore.
And if the team is moving TOO fast for you to build any fury, I'd say don't even worry about it since the team is steamrolling anyway. 
Are there any tricks to doing this? I have this problem on my fifty /ELA and my new shielder. It seems like no matter how fast I move, unless I take the alpha strike (which often kills both characters) I can never maintain any fury above 30-40% on teams and on the times I do fill it up (which are rare) by the time the lot of people move to the next spawn it's almost completely gone.
Virtue:
Miserya - 50 EM/ELA Brute (Perma-shelved)
Adriana Rayne - 42 Katana/Dark Scrapper
Cyberpulse - 26 Super Strength/Willpower Brute
Steel Heart - 24 Invuln/Super Strength Tanker