With "drama" they're seeking the deep intimacy of conflict: when you are in serious conflict with someone, you have stripped away all the social norms that you usually hide your feelings behind, and you're showing your true self. If there isn't someone in your life who is causing conflict with you, and you crave intimacy, you might try to create it by convincing yourself that there IS conflict. Unfortunately, if they won't play along and "fight back" you don't ever get to a point of sharing true feelings, because the situation has not been completely created. To get conflict intimacy you need two (or more) people invested in settling a conflict: if the other person is convinced that the conflict exists only within you, they won't invest, and the situation will not be created for you to both share deeply.
With sex, they're seeking the deep intimacy of learning another person's desires and pleasures and sharing their own, and meeting them in a space beyond the physical. For people who aren't conscious of this as the process, they think it only happens accidentally, and they don't know how to produce that intimacy. Since it happens most often at the beginning of sexual interactions, before you create your set of assumptions (and thus are still open to learning), those people will often seek new people and/or new experiences to try and stumble upon that intimacy again. Obviously sex can be sought for the physical, and novelty for its own sake: but they can also be sought as a very inefficient way of getting intimacy.
These intimacies are just little flecks, gained at the cost of losing people or spending lots of energy. It's so important to know what you are really seeking, and to seek it consciously, because people in need of intimacy can be voracious and consume others, creating harm in themselves and those around them.
To create intimacy in a steady, nourishing way, you have to do it on purpose. You have to find someone who already practices openness and honesty who wants intimacy -- not just sex or romance or fun or an activity partner -- and practice. This means when you find something that makes you BOTH feel intimate, you take note and do it regularly, as a commitment to each other. If the person who gives you butterflies or seems "just so awesome" does not want -- ON THEIR OWN and not as a response to a request -- to be intimate with you, find someone else. It is hard to find people who want to actively practice intimacy, but if you want to actually find it you have to find self-motivated people, not try to convince someone that it's a good idea. They also have to be self-motivated for honesty and openness. AND so do you. If you want to keep your image or hide things, you are restricting yourself to accidental moments of intimacy rather than deep, continuous intimacy. You will not get intimacy while trying to control how you are seen by others; that's like trying to make out with someone while wearing a mask that covers your mouth.
The ways that have worked for me, after finding a self-motivated partner, are these:
1) bringing up small upsets or discomforts immediately. Not only does this prevent them from turning into huge resentments that hurt everyone, but it gives small amounts of conflict intimacy. Handling conflict with kindness can be every bit as intimate as the most intense sex.
2) talking regularly (I prefer daily) about all the emotionally impactful things in your life: including and ESPECIALLY small impacts. If it's something good, it's pleasure-sharing intimacy like sex is. If it's something bad, it's conflict intimacy like an argument -- with no risks to your relationship. If it's neutral, it's not usually emotionally significant, but if it is, it provides history for greater understanding. Don't just share the ones that go with the image you like: share the ones that make you "look bad" or "sound bad" too. We all have evil bits and sharing them is an excellent way of working through them. also, see #1. Your relationships with people who mean a lot to you are full of emotionally significant experiences you can share, whether those experiences are past or present.
3) sharing experiences that are emotionally significant. This sounds simple but there are many ways to do it that don't often get considered. Going to significant places (like your childhood home, places you had "firsts" or spiritually significant places) and/or sharing media that has deep meaning to you (like watching a movie you resonate with and explaining why, or reading each other's beloved books). Also, anything that brings you great pleasure or creates conflict within you will be intimate if shared: if you both really love roller coasters, that's emotionally significant. If you both really hate something (like prejudice or pollution ;-)), work against it together.
4) creating together by cooking or painting or whatever: it's intimate like making a baby, but with less mess and stress.
5) asking questions. It makes it easier for the other person to share, and it helps you to think more carefully about what they say.
6) silent prolonged eye contact. This is a big damn deal; it's hard, but it's so good.
7) focused, undistracted touch. I think this is part of the reason people like getting tattoos and getting their hair done: it's a completely focused touch. Casually holding someone's hand doesn't offer the same intimacy as holding someone's hand, gazing at it and stroking it. Massages and grooming each other also work. Sex can fit into this category, but often doesn't because desire is very distracting. I think that sensation play and BDSM play offer this intimacy. For it to be undistracted you have to take turns!
8) sharing spiritual and/or altered-state experiences. I know this doesn't work for everyone, but if you DO have a belief in spiritual things and/or if you use mind-altering drugs, sharing them is intimacy. Talking about experiences is intimate and sharing experiences is even more so. This sort of reiterates #2 and #3, but I think it's worth its own number because (for me at least) that is harder and more rewarding than most shared topics or actions. And you don't have to have the same beliefs for the talking to be intimate. If they're not far off from each other or you're quite creative you can make shared experiences as well.
9) sex, IF you're sharing things that feel vulnerable or spiritual, and you're thinking about what you're doing.
10) silliness! laughter yoga and other playful things that you'd be embarrassed to do in front of most people are excellent for creating intimacy.
I don't expect that all of my methods will work for everyone, but some might: and if you have others I haven't listed, please share!