OP - I find it quite stressful reading your posts to be honest. That's not a criticism, it's just stressful to see someone putting up with so much bullshit.
CAREER ADVISOR - Were you actually THERE when the career advisor allegedly said this? Because I have a relative who works as a career advisor, and this just ISN'T the type of thing they say. Which makes me strongly believe that your husband is lying through his teeth. A career advisor simply wouldn't say something like that.
Don't move house to be closer to your husband's work. You need to keep working, so that you aren't financially independent on this man. Sorry, but it's true. You need to keep your options open in case you decide you can't deal with this anymore. Also, I rather suspect that if you became a STAHM, you might find that he starts travelling even more, as he will have a full time babysitter.
I'm angry on your behalf that your husband is treating you like this. Doing a job that requires lots of travel - okay for a short period of time. Okay whilst you establish yourself. NOT OKAY to be doing for years on end, ESPECIALLY when you know that your wife is struggling.
To be honest, I think this situation depends on you. You sound way less angry than I would be in your situation, so maybe this is just something that frustrates you every now and then, but in reality isn't that big of a deal to you. In which case your husband is never going to change jobs, because he has no motivation to. I've travelled quite a lot with previous jobs. Don't think for a second that he isn't having a good time while away. Yes the meetings can be tough, but you stay in nice hotels, you have drinks with colleagues, nice dinner's (usually paid for by the hosts or your company), and it's generally great. If it wasn't fun, he wouldn't still be doing it would he.
If you really aren't THAT bothered by his selfishness, then you just have to accept this is how it is, and the situation will stay the same until the day that HE decides he can't be arsed with it. Until then you will be burdened with the realities of childcare, running the home and picking up the slack.
However, if this is something that GENUINELY bothers you, then you need to lay your cards on the table, and actually stick to what you say.
Prime Minister and Royal family aside, I can't think of a single job the requires this much travel across the board. Fuck it, I've worked with sodding SAILORS who gave up sailing and started working as ship inspectors, or legislation writers, or at the IMO (international maritime organisation based in London). I've known people work a few years in humanitarian response or disaster recovery, then move into stable jobs in the foreign office, in charities, or in regional offices of UNESCO. I've known friggin' PILOTS who have moved into positions at big airlines.
There are always, always other options.
So if you have genuinely had enough, you need to tell him that you are fed up of picking up the slack, and you aren't prepared to spend your life in an uneven partnership. Either he speaks to work and drastically reduces his travel, he finds another position which doesn't involve so much travel, or you separate. Tell him he has six months, and if you can't see that he is making an effort to change, then it's over.
If after 6 months nothing has changed, separate. If after six months he is clearly making an effort but not quite there, tell him you will give him another six months. If after a year it's still the same, be true to your word and separate.
Let HIM realise that children bring responsibilities and he can't just fuck off for weeks and weeks during the year and abandon his family.