So I want to dig it up, lower the grade, and put in a drainage channel. The problem is the area to be dug up is possibly an easement on my neighbor's property.
Not a lawyer, but I think you need to find out the nature of the easement.
Most easements give a local authority (utility, HOA, city) access for specific purposes, and the land is otherwise private.
I have never heard of an easement that would give a neighbor the right to privately dig/install drainage across the property line.
It's pretty common for driveways (for long, narrow country lots, or shared driveways) and lake access.
OP, how is your relationship with your neighbor? Required or not, it certainly would be neighborly to let him know you have a problem that needs to be addressed, and it may inconvenience him. I would hope that the first he hears about it is not when official paperwork comes in the mail. They may even have some background on why / what has been done before to try and address it.
Sure, but that's more like an access easement. Roomtempmayo makes a good point, most easements have a particular purpose, all have a particular easement owner. I worked with a public utility, and if they had an easement that read "Water utility" but they wanted to repurpose it for a sewer installation, they had to go back and sign additional paperwork with the landowner.
OP needs to figure out what type of easement exists, and who owns it. Being OP has a zero lot line, it wouldn't surprise me to find an easement that covers only "house maintenance use" things like access for pressure washing walls, painting walls, replacing siding. If there is a drainage easement, it likely belongs to the HOA for purposes of moving water from one place to another. In that case, OP would need the HOA to do the drainage design, permitting, and installation.
To answer OP's confusion about "maintaining the easement" this refers to normal stuff like mowing grass and preventing shrubs from growing up in the easement area. One cannot interfere with the purpose of an easement, by digging a pool overtop of a water line, dumping a load of soil into a drainage ditch, filling a stormwater line with concrete like a crazy homeowner did in Florida.