In contract law, for a contract to be binding, the parties have to both have a "meeting of the minds."
By that, the law means that both sides have to enter into the agreement, both understanding the terms that they are entering into and both agreeing to those terms. The employer here can certainly argue that they did not intend to enter into an agreement for the second 26 week severance and you too can argue that they did.
However, if you never received a countersigned document from them, that will be a factor that works in their favor, because they can then state that they never actually entered into the agreement. Of course, you can argue that they entered into the agreement by their actions, but what would their actions show? It would show that they accepted the agreement at the lower rate, not the higher rate, at which point they'd have the stronger position in that argument too.
This would be a very academic case, because it involves a lot of questions about what was actually agreed to, did both parties agree, what actions did they take, etc?
Because severance is not legally required, it is completely possible that the court would find that no agreement was reached and you would get no severance, while the employer would lose out on your waiver against suit (as contained in the original agreement you signed). Furthermore, it would end up costing you the hourly rate of an attorney to sue for breach of contract, because attorneys don't take those on a contingency fee basis normally.
On the facts you've outlined, unless you got a countersigned document from them (at the original rate), I would not consider it worth the expense of challenging because you'd probably lose (at least, the facts are more in line with their position). If you do have a countersigned document from them with the original rate, you would have a more reasonable argument, but you have to weight the potential gain (the difference in the severances) against the cost of suing in state court at an hourly rate, something that could reasonably cost $10,000.