I don't think you can extrapolate from centenarians. As Jimmy Stewart once said about age, "after 70 it's patch, patch, patch." I don't know anybody that has made it to 100, but several in their 80's and a few in their 90's, but these people were invariably fragile, and most of them were a bit mentally addled as well. Not one of them could run, for example. All had eyesight problems, mobility problems, and most had memory problems. And though I did not know anything of their sex lives, I truly doubt there was anything to know.
And due to their physical deterioration, those people were all pretty conscious of their impending demise, and alluded to it more often than young healthy people do. If nothing else, in their preference to review the past and things they lived through, rather than speak of the present or the future (in politics or entertainment or plans of the younger people they know). They know they've lived their lives, basically.
A person living to 300 would have to be pretty healthy for most of it, I don't think a story about somebody living 200 years as physically and mentally degenerate as a centenarian, being cared for and living in the past, would be that good.
I would presume that their "elderly stage" is no different than anybody else's: 20 or 30 years, and the same for their "childhood" stage. The human brain is not actually fully developed into an adult brain until the age of 24, give or take two years. Make it the same for your long-lifers. Put all the extension into the middle, and scale it: So between the ages of 25 and 275, they are like people between the ages of 25 and 75; and they age proportionately: That would be 250 years versus 50 years; so a factor of five.
That would hardly be noticeable at first; at 30 they look and feel like a 26 year old. But by the age of 75 (50 years older for normal people) they look like 30 year olds, and that will stand out. By age 125 they look like 40 year olds, etc; for every 50 years they only seem to age 10.
Their psychology, if they know this or figure it out, will change significantly toward a longer view. I'd expect them to invest more time in education, for example, because in the long run it costs them ten times less in terms of life years. If they have children, they are much more likely to become peers with them, they will look and act the same age and won't always be the "older and wiser", a 90 year old kid that looks and feels 31 isn't going to defer to their 110 year old parent that looks and feels 33; they will be far more equal.
These people are likely to engage in serial careerism, as well. If they are intelligent, I wouldn't be surprised if many became business types, and professors, and teachers, and lawyers, AND medical doctors, perhaps investors, entrepreneurs and politicians as well. Architects, engineers, astronomers, actors, writers, whatever they feel interested in, unlike us, they can pursue as a full career.
I mean MANY of those careers for a single person; because our society is geared to these careers lasting roughly 25-40 years from education to retirement, and these people have about 260 years (20->280) from education to retirement. If they hurry, they can fit in TEN of what we would consider "life-long" careers.
Finally, their psychology of lost loves (and perhaps family, if their children are not guaranteed to be long-lived) will have to change as well. For RL centenarians, everybody grows older at the same rate as they do. They lose friends and lovers, even children and grandchildren, but they never experience the reversal of somebody younger than they are, aging to be much older (in appearance) than they are.
These long-lifers might form life-long friendships amongst themselves, but amongst normal people are likely to consciously refrain from such commitments. Instead they are likely to "keep moving", making friends within a few years of their apparent age, staying a decade and moving on. I am not talking about any legal or identity ramifications of them not aging, there doesn't even have to be any. I am just talking about the emotional ramifications: When your best friend has aged twenty five years and you have aged only five, the difference is like a modern person having a best friend twenty years older, old enough to be their parent. 40 v. 20, or 50 v. 30. And for the long-lived person, how about the age difference after 50 years; when their friend is retired and in a nursing home, and they have aged from apparent 25 to apparent 35?
It's too much, in terms of energy levels, culture, aspirations and plans, love life, life interests, and everything else. It is better to break it off and start over, in a new town, or even a new country. That might be emotionally difficult, but might actually be better for both, to keep friends with aligned interests for their stage of life. For the same reason, I'd expect them to refrain from forming any deep emotional attachments to short-lived friends, i.e. to intentionally shy away form "best-friend" bonds at all, and maintain just co-worker and neighbor types of friends, bonds that can be easily broken with a few "it's been great" and "I'll miss you" and a "bon voyage" party. Get out of their lives without breaking their heart.
I'd say the same thing when it comes to dating. Either they need to "come out" to a love interest about their long life, or keep their relationships short, so they don't break hearts and ruin lives. (Assuming they are emotionally normal and care about other people, and aren't sociopaths.)
I imagine psychologically, this could be a relatively solitary life. If there are many of these long-lived people, they would self-segregate to form a virtual community for friends and romance. Especially if the trait is heritable; so parents do not risk outliving their children, seeing them grow from infants to decrepit elderly patients and die. I'm not sure but I think if I knew that was almost certain to happen, I would refrain from becoming a parent at all.