HI. PDtax here. I wouldn't normally respond to further muddy the responses, but I think your follow up question did not get answered. I will respond, then opt out.
When a divorce decree is issued, it is binding on the parties, but IRS does not have to be bound by that agreement. Your assumption of any tax liability is not binding on IRS, so they can pursue one or both parties to collect.
General advice about innocent spouse relief and the like, in my opinion, does not answer your question regarding the interplay between divorce and joint tax liabilities
. If your agreement with your ex-wife states you will be solely responsible for the tax debt, and you settle, IRS can still look to her. The language of your agreement may come in to play, but such an agreement is not binding on IRS. They could look to her, and she could look to you for relief. Think additional court proceedings.
If you are indeed an Offer candidate, you might want to consider a Chapter 13 filing instead of an Offer. Depending on the ages of the taxes, you may be able to pay a portion of the tax debt, and since the bankruptcy court would have the authority to supersede your divorce agreement, you might find that to be a better course of action.
You didn't provide facts to make that call, but I would recommend a consult with a quality bankruptcy attorney, and/or a bankruptcy CPA to assist in the review of your options.
For example, your joint 2009 tax return, if filed April 2010, could be eligible for bankruptcy relief (it's more than three years old).
Because there have been multiple answers to your question, I will opt out. If you want anything more from me, please ask for PDtax in the subject line of a new question, and the other Experts should respect that request.