All of your problems are with the roommate, not the landlord. While others have said a reasonable landlord would work with you, I can think of one big reason a reasonable landlord wouldn’t work with you. If the landlord doesn’t think he’ll be able to get his money out of the other guy (damages and the balance of the rent), he’s not going to let you off the hook very easily.
Speaking of damage… you’re probably not liable to the landlord for half the damages, you’re probably liable for all the damages, plus all the rent. The lease probably states something along the lines of you each being totally responsible. If your roommate doesn’t pay his half, the landlord can come after you. It would then be up to you to go after the roommate. If you move out without getting yourself off the lease, the roommate could stop paying rent, trash the house, get evicted, and ring up a huge bill for the whole mess - and you could be responsible for the whole thing.
If the two of you moved out, the landlord would have the responsibility to attempt to rent the house back out to minimize the amount of rent you would have to pay. I’m not so sure he has that same responsibility to find your roommate a new roommate if you move out only. I doubt it.
Are the dogs a violation of the lease? If they are, don’t be surprised if the landlord starts threatening to evict both of you.
If the you and your roommate were neighbors in an apartment building, and your roommate was causing problems and the landlord wasn’t doing anything about it, you’d have cause to break your lease because the landlord has the responsibilty to provide a safe, habitable environment. But, since you two are roommates in the same house, you both are responsible for what goes on in the house.
Have you talked to your roommate about wanting out? Maybe his girlfriend would move in and take over your half of the lease? Whatever you do, don’t move without the landlord officially removing you from the lease, because regardless of what agreement you might work out with your roommate, the landlord can come after you as long as you’re on the lease. You would have to sue your roommate to get anything you paid to the landlord back.
What’s the income situation with you and your roommate? If the landlord was to evict your roommate (presumably for the dogs) and let you stay, can you manage the rent on your own, at least until you find another roommate? Can your roommate handle the rent on his own? If so, the landlord would be more likely to let you out.
Your first step should be to talk to your roommate. Maybe his girlfriend would move in. Maybe he knows someone else. Maybe he’d just be happy to get rid of you and would pay all the rent himself. If you can work out something with him, going to the landlord to make it official will be a lot easier.
If that doesn’t work, go to the landlord and explain your situation. If he does agree to let you out, make sure you get something in writing. Also make sure you have a total list of what you’re expected to pay in damages and rent, in writing, before you leave.
Now, if you can’t work things out with the landlord, be prepared to move anyway. The landlord might decide to start eviction proceedings against the both of you if he has reason to believe he’s not going to get his money.
Your other option would be to try to evict your roommate yourself. I’m not really sure what rights you might have there. You might also be able to sue your roommate under some sort of breech of agreement to force him to take responibility for the entire remainder of the lease. I’m not a lawyer, so I’m just guessing here.
One last possibility, if you ever wanted to be on TV, you could take this whole mess to People’s Court.