Why is there rust on the gun to begin with? You do realize that a firearm must be periodically maintained even when not used, otherwise it can rust.
I'm not really sure what part you're referring to when you say "the dowel." Are you talking about the ramrod? I'm not familiar with the New Englander specifically, but I suspect you're talking about the part under the barrel that the ramrod is held against. If so, I'm pretty sure that if you broke it off, you're going to have to take it to a gunsmith or send it to T/C to be repaired. As for the breech, if the breech is like my White Mtn Carbine, then there is no removable breechplug, just a hook like stud that extends from the back of the barrel to secure it to the stock.
Take so pictures detailing what you were doing and what you (think) you broke, and we might be able to help you better.
Mike