I think your wife needs to know this affects you. Disjointed schedules are killers in a relationship. I know I live in one. You have to find a "safe" babysitter your wife FULLY trusts.
The issue is not to look needy, but she needs to be made aware that quality time alone with her spouse is critical even when in full on mommy mode. Tread carefully though.
The babysitter at home is a great idea... also are there any neighbors you could befriend and take turns watching each other kids. When my kids were young we had an older couple neighbor across the street basically be in essence another set of grandparents to our boys...Oma and Opa. You would be surprised at the kind hearted people around you. Neighbor babysitters are the best! Built in trust factor too.
Here's a good line to use on your wife in mommy mode...
"Honey is there anything I can do for you today to make your day easier?"
They eat that stuff up... Good Luck.
Also try to create "Experiences" for her at home... dates don't have to be out. Maybe invite her friends over as a surprise (You babysit). Or tell her you need 20 minutes to try a new dance move with her etc. Have a big family movie night with all the trimmings etc. There are lots of dates you can set up in your home. Women want "experiences" they also want to see you nurturing the kids. Create experiences involving he whole family... you plan it. Your wife will want to reciprocate eventually.
In the end though you have to realize she is a mom now... kids will ALWAYS come first from here on out and it only gets worse later. I was having great sex at your stage... now two years sexless (kids teens) BUT we are almost past that now. Strap yourself in the ride just began for you... hold on tight and don't fall off. You are a dad! Hardest job in the world and you get scraps of support. Your kids can screw up and your wife loves them no matter what. You don't get that same treatment...sorry. Tread carefully everything you do could be another resentment (everything!)... its like walking a minefield. I wouldn't trade the job for anything though well worth all the hassle. Its crazy but that's the way it works. Have to stay strong. Good luck..you'll need it! We got an awesome dog (G. Shep.) that helped. I miss my dog now
. Hard losing your dog and wife sexually and your job at the same time.
At least I found an awesome job... now I need my wife back (almost there).