On "Jo Writes Stuff", Jo has produced an epic analysis of whether or not a character is a "strong female character"; and a test to go with it. Here is her instructions on How To Use The Test.
She has stopped any new analysis, but here is a list of All The Characters She Reviewed.
I believe this can help you with some of your issues; just writing a post-woke female character.
For the most part, other than sexual attractions or attentions, I think in the 23rd century the answer is to write female characters and male characters rather similarly, but NOT as women becoming like men: More like meeting in the middle, or both sides going to some extreme.
For example, consider makeup: One route is to eliminate it; women no longer need it in the post-woke society. But the other route is to make it universal: Your male character may have some trouble in our modern time, because like all men of his time, he doesn't usually go out without makeup. (The same was true for men in some royal courts a few centuries in our past).
Your guy might have acquired many habits and attitudes we might consider "feminine", and what we consider acceptably "masculine" today may come to be considered just rude in the 23rd century. Like men interrupting women, or clerks and managers speaking exclusively to a man instead of to both the man and woman.
-- Edit in response to comment:
Men interrupt women 2-3 times more often than they interrupt men. Women interrupt men and women equally, and less often. See This academic study specifically on conversational interruptions (results are on journal page 430), and see popular articles Why men are prone to interrupting women in "Women in the World" 03/09/15 and The Universal Phenomenon of Men Interrupting Women in The New York Times 06/14/2017.
The phenomenon of men that need to address a mixed gender group (or couple) beginning their conversation with a male of the group is similar; male gender is treated (on average of course) by men throughout society as a mark of higher status / power; almost regardless of actual status / power. To be clear, they don't refuse to speak to a woman, it is just that in a mixed-gender group, they generally choose a male in the group as the "leader" to speak to when addressing the group, far more often than they will choose a female within a mixed group.
This is a form of status bigotry, which post-woke society would likely not engage in: They would use other gender-less cues to choose a "leader", like some combination of age, grooming and bearing, or they would ask to speak to somebody about X, and talk to the person that stepped forward.
Back to interruptions: Female politicians, business leaders and judges are interrupted more often than their male counterparts. Again this is a "bro-culture", women attempt to interrupt both men and women equally; but have fewer successful interruptions with men: The men being interrupted are more verbally aggressive against women that attempt to interrupt, and men in a group are more supportive of that aggression against a women interrupting, than they are if another man had interrupted.
-- End of Edit --
Or on the flip side, your male may be just as motherly and devoted to kids, and know everything there is to know about childcare and raising children, and doesn't defer to women in the least in that respect. In his time, men are expected (by both other men and women) to be full and sensitive participants in childcare.
For the most part, I would just have your 23rd century people not perceive any difference between interacting with men or women. As a psychological trick to help you do that and get past your own cultural bias, she responds to and speaks to men as a modern woman would respond to and speak to another modern female stranger, saying or doing the same thing: Gender makes no difference to your time traveler. The opposite trick for your time traveling man: he responds to and speaks to women the same as a modern man would respond to and speak to another modern male stranger.
And though they are a couple, they engage with each other as friends. If you intend to have romantic or sexual scenes in the story, make them equally desirous.
There is seldom a need for designing a "strong male character", But I'd use Jo's checklist in the same way. Here are her questions; the link above has details on what you should consider for each question.
- Does the character shape her own destiny? Does she actively try to
change her situation and if not, why not?
- Does she have her own goals, beliefs and hobbies? Did she come up
with them on her own?
- Is her character consistent? Do her personality or skills change as
the plot demands?
- Can you describe her in one short sentence without mentioning her
love life, her physical appearance, or the words
‘strong female character’?
- Does she make decisions that aren’t influenced by her love life?
- Does she develop over the course of the story?
- Does she have a weakness?
- Does she influence the plot without getting captured or killed?
- How does she relate to stereotypes about gender?
- How does she relate to other female characters?