This is not that uncommon. Some points: (I did not have a regular location that I travelled to, but lots of away from home work travel with kids).
1) Kids under 6 years don't really understand it. My 3 y.o. slapped my face and told me not to do it again after my second 2-wk trip away (gone 4 our of 5 weeks). I then chose to cut drastically back on overnight trips and ended up with a lot of 20hour work / travel days. Kid was happy knowing that mommy was coming home after he went to sleep.
2) After they got older, you do lose touch with homework, interests, etc. It is a bit hard to fully turn off and turn on for the family in the week you are home because you are used to relaxing in a different way than around people.
3) The thing that drained me, however was two-fold -- working from home / local weeks -- my team was distant, and eventually I reduced the travel to about 1 week a month, and having a distant team makes you feel a bit distant, too. AND...
4) i really hated never being able to sign up for an exercise class, PTA, weekly meetup hobbies, volunteer commitments. I rarely entertained when we were home because I was trying to fit in so much family time or my schedule was erratic. Because I had differing destinations, I could only make dr / dentist and hair appt at my home location, so pretty much stopped doing that because my time at home was too short to take off from family to spend the saturday going to the medical clinic or dentist (ok, I did start to randomly go to different hair places, but the other appointments stopped, I have almost never taken work off to go to a "routine" medical appointment)
I did make a lot of money by being able to focus on the career, though. I could have gone to bigger and better promotions because of it. Instead I FIRED last year. Now I have TONS of time with my teenagers.
My two cents -- try it out for 8 months. A lot of companies change what they want from you about 6 months after a merger, and you may be able to negotiate a new "close to you" office, or even that 2 hr commute one at high pay (and you rent a room close to work that you stay at 2 nights a week and live / work from home the other 4 days).