Why would this need to be a miltary organization? Are the sinking lands/rising seas a plot from some enemy? This is a civil engineering thing if ever I saw one. Might want to look at how the Dutch deal with this. I don't think it involves any armies.
In case you missed it, that was a joke. There is infact an Army Corps of Engineers that is responsible for exactly what the comment suggested (amongst a lot of other things).
The Army Corps of Engineers has its prominent civil engineering role because early America did not have a lot of federal resources and was born from war. So when the Federal government decided it wanted to take on large scale civil engineering works, the only ready to go resource at hand were the military engineers. And then afterwards, it's pretty much been inertia.
The Army Corps of Engineers civil works division is basically almost completely staffed by civilians. So there's a convoluted top level organization, but on the ground, it's not like they have soldiers and military engineers building levees.
Congressional Research Service report: https://www.congress.gov/crs-product/R48322
It includes a section about discussions on transferring civil works responsibility out of DoD.
Because if it were civil and paid for for centrally, it'd be Big C Communism marching in. Let the army do it and it's eagle-riding patriotism.
But also if you do declare some sort of emergency that allows this, otherwise frustrating checks and committees can be bypassed. Probably not a bad thing.
The way the Dutch deal with this is changing. The predominant view now is depolderization. Environmental concerns indicate that they should return the land to the sea.