Do the classes feel balanced?

No the classes are not balanced but you have exactly the opposite idea of the issues as most everyone else.

Assault is pretty widely considered the single best class in the game. Their primary weapons have some the best accuracy and lowest time to kills of all the classes. The vanguard shot gun skill is the most overpowered skill in the game bar none. The vanguard shot gun is capable of wiping out entire squads of enemies, and taking out any reinforcement type that isn't a spaceship. It is the best tool available to any standard troop for killing heroes, vehicles, and elite troops. The self healing ability available to the assault grants more durability to the assault than any other ability basic units have except for the command buff that officers have. Assault is by far the most flexible and powerful overall basic troop in the game.

Heavy on the other hand is THE WEAKEST troop in the game when it comes to infantry to infantry combat. Heavy's primary weapons have the lowest dps and highest spread of all the primary weapons while requiring the highest amount of time on target of any of the primary weapons. Unlike every other class's defensive abilities the heavy suffers enormous mobility penalties for theirs. They also suffer that same enormous mobility penalty for their dps boosting skill. The only saving grace of the heavy class is his solid tool kit against vehicles, especially objective vehicles such as MTT and AT-AT walkers. 

Officer is widely considered the second best class, though on certain maps they can even outperform the assault (maps where the assault have trouble using vanguard effectively like outside on Hoth). They can out-snipe specialists. They can support their teammates which is especially effective when paired with reinforcement troops. They have the best defensive and disabling skills of any class. They are way more effective at playing the role of a squad anchor/enabler than the heavy is.

Then there is the specialist. If you are a head-shot deity then go ahead and play specialist. If you aren't able to hold a bead on heads with exceptional talent, you're better off playing something else. Specialists have the least tools against reinforcements. They have the lowest health pool. And with infiltration being nerfed their ability to be effective on the front line and actually provide support to their team has been greatly hindered. If you are a great sniper, you can still dominate with specialist, but overall I would say that specialist is the weakest overall troop. They have a solid primary weapon but the rest of their kit pales in comparison to what other troops get.

I would submit to you that if you are playing assault and having difficult against heavies that you aren't playing the assault class correctly. You should use more team play and just team shoot heavies. In a team setting the heavy is practically useless, which is my biggest complaint about the class. If you aren't using squad tactics you should be flanking. The vanguard shotgun will 2 shot a heavy with his shield up in less time that it will take that heavy to kill you and one shot a heavy if you catch him by surprise or without the shield available. From range the assault's weapons are way more accurate. Use cover and you have a clear advantage over a heavy.