We don't date much (maybe go out once a month; no close family, few available, trustworthy babysitters)
It's ok. Most people who come here are full of excuses, too, lol.
Seriously? Few trustworthy babysitters? How would you know? What steps have you taken to find them? Agencies? Churches? Friends' recommendations? Angies list? Care.com? nextdoor.com? A dozen other websites?
she complains about the lack of romance I show her. If romance is picking her up and carrying her to bed it's usually too late by the time there is any opportunity
More excuses. And you haven't even started implementing our advice yet; of course you can't just carry her to the bedroom right now. You're a doormat who frankly disgusts her right now. You have a LOT of work to do first before she'll even remember she used to be attracted to you, let alone care about being attracted to you. Read NMMNG before you do ANYTHING. You will understand.
If she complains about the lack of romance you show her, what are you doing about it? What does romance look like to her? You should know that, you're married to her. Or do you two never talk to each other?
and I've been rejected no matter how manly or tough I get (the toughness always offends her).
What are you talking about? What toughness? What does that even mean? Are you talking about during sex? You realize that is NOT what we're talking about, right? A strong man never has to use force. Never has to be mean or tough. Never raises his voice. Because he is SECURE in his manhood. In his desirability. In his ability to weather any storm. In his lack of need to get something from her, because he's complete with or without a woman.
At night she's usually so exhausted from her day that after dinner she just wants to binge TV and fall asleep (I usually make sure the kids are washed, brushed, read-to and tucked in at night and she does the morning wake up kids, dress them, feed them, take them to school, pick them up every weekday).
Isn't she a SAHM? What the hell does she have to be exhausted about, since you're the one doing all the work from the time you get home from work? And then giving her footrubs every other night? And backrubs?
How fast CAN you read NMMNG?
How about when she just wants to binge TV and fall asleep, instead YOU sit down on the couch and ask her to give the kids their baths? The fact that YOU are the one working full time and THEN going home and working all night while she has every day off from 7 to 3 IS YOUR WHOLE PROBLEM.
Do you see that yet?
The only romance that ever stuck with me was writing and touching, but the inspiration to write is blocked and the touching is working against me.
More doormat talk. Read the book.