I think you will be o.k.
And I have 100,000 miles experience.
Plus, I am not as young or as thin as I used to be.

You have kids and a job - not like you can go out there and train 24x7.
And I suspect you also have a lawn to mow, soccer leagues to attend, etc. etc.

You are aware that the Paw Paw tunnel is closed - probably until Aug/Sept.
You should take the road detour rather than the killer trail over the mountain.
Late June is o.k. - it gives the C&O portion a bit of summer weather to dry out.
But be aware that heavy rains can make the C&O a tough slog. (Less so the GAP)

Even if you haven't done a lot of riding beforehand, 3x commutes a week should be o.k.
And it sounds as if you will be traveling light. Your itinerary looks quite reasonable.
Be aware that fixed stops are a bit tougher than a flexible itinerary.

Just plan not to overdo it. 60 miles the first two days = 4 x 15 miles.
Start early - ride for a 1 1/2 hours - then take a long stop - then another 1 1/2 hours to lunch.
Take a long lunch and maybe a nap. Then two more legs in the afternoon.
There is no one who is timing you. If you leave at 7 a.m. and arrive at 7 p.m., who cares?
You can stop often and explore all the interesting places along the way.

If you like to read, take a paperback - you will have 100% reception with a paperback.
No one says you can't sit on the riverbank in the shade and read for an hour.

Will, I am a believer in doing something now rather than later.
Because, quite frequently, "later" never comes.

Wishing you a wonderful tour - Jama