The short answer to your question is "yes, of course." Two societies that come into contact, assuming that they are able to communicate, will necessarily have an exchange: communication requires this. To put it in Star Trek terms, the Prime Directive is idealistic but utterly impossible: the only way not to influence another society is not to let them know you exist.
On the other hand, there is no reason that one has to think of such contact in catastrophic terms. To be sure, the history of European colonialism is not a pleasant one, but that needn't be the sole model.
Consider, for example, the many, many tribal peoples of the Americas, prior to the arrival of European invaders in the wake of Columbus (let's ignore the Vikings here). They talked to each other, traded with each other, sometimes fought and sometimes didn't, for an extremely long time. As a result, we can spot remarkable affinities among myth-cycles all across the Americas. If we had better longitudinal data than we do, we could chart how customs and such moved; in some cases, we can do this to a degree, especially when it comes to material culture (pottery, beadwork, etc.).
You cannot possibly model all of this, of course: it's ludicrously complex. But there are a few points that could perhaps be useful in a fictional worldbuilding context.
1. Variable Usage and Interpretation
Suppose one culture has a big thing about hats, where there are all these incredibly fine variations and gradations in hat shape, style, material, color, and so on, and these are a complex sort of code having to do with social station, age, sex, place of origin and/or residence, and so forth. (At various times China has had some of this kind of thing going on.)
Now along comes the other culture, and they do some peaceful trading for a while, and this other culture takes home pictures and souvenirs and stuff in addition to actual trade goods. Now at home, they might end up with a big fad for "alien hats," and suddenly the millinery industry is going through the roof. Of course, this culture either doesn't understand the hat-code system, or doesn't understand it well, or just as likely, sort of understands it but doesn't really care: the alien code doesn't apply, but the hats look cool.
But maybe that second culture, after a brief fad about the alien hats, finds itself in a situation in which everyone is wearing weird hats all the time. They have the concept of the alien social hat-code, but it's an alien system that doesn't match their own life-patterns. It could well happen that, over time, this society develops a way of coding hats that does match their life-patterns, particularly if they're constantly trading with the original hat-wearing aliens.
3. Rejection as Adaptation
And then a new thing sweeps our society: hat-coding is an alien thing, and it's contrary to what we're really all about (because of the gods, or pride, or whatever). So people stop wearing hats entirely. Before alien contact, they sometimes did and sometimes didn't wear hats, but it didn't mean much: it was a fashion thing, or maybe a way to keep the rain and sun off. Then there was this fashion surge, and then this new hat-code, and now nobody ever wears hats.
I realize that this may seem like a silly set of linked examples, but the point I'm trying to make is that cultural contact and adaptation is both extremely complicated and to a significant degree non-catastrophic. While it is obviously possible for "contact" to mean "convert them at swordpoint and then enslave them," it usually doesn't work that way. Usually what happens is that objects, customs, images, and so forth are exchanged, and then each culture has to make sense of the new stuff in its own terms, which in turn changes that culture -- but not necessarily to make it any more like the one it's exchanging with.
Here's a final example, in this case from real history: Melanesian cargo cults.
Cargo cults are (mostly were) complex and varied, but the basic phenomenon is revealing and helpful for the question at stake. To give a highly simplistic account of one type: During the Pacific War, the American navy (especially) sometimes provided various goods to local Melanesian tribes. In essence, the idea was to get these people to be friendly to Americans, such that they could be enlisted to assist against the Japanese; for example, they might tolerate an airstrip, or they might tell Americans about spotting Japanese ship movements, or whatever. Now when the war ended, so, pretty soon, did the providing of free stuff.
Now the problem is, a lot of Melanesian societies base their economic systems in social exchange. Think of it sort of like The Godfather: if I'm a big powerful dude, that's not because I have a lot of goods, but because everyone owes me favors. And to retain and increase that power, I have to keep people owing me by giving them stuff and not letting them pay me back. For my birthday, I'm going to throw a huge party and invite everyone, and I'm not only going to feed them but I'm going to give them all presents. This both demonstrates and increases my power.
So for these Melanesians, the American goodies represented a problem. The Americans gave them stuff, but didn't accept anything back. So this meant that the Americans were incredibly powerful. But when they just stopped giving stuff, what were the Melanesians supposed to do? They owed favors, but now the folks to whom they owed the favors wouldn't talk to them. So in these cargo cults, you'd get a millenarian religious movement centering around the Americans and their stuff. People built wicker airplanes, put on wooden headsets, and generally did anything they could to force the American "gods" (loosely speaking) to restore the relationship, not necessarily because the Melanesians wanted or needed the stuff as such, but because this gross absence threatened the whole system of exchange itself.
If you do a little research on cargo cults, and think about what's happening on the two ends of the exchange, you'll end up with a very wide range of ways to think about cultural adaptation.