I think they beat way is to bring the subject up again and ask why they feel cautious (awkward) around you and ask if you did or said anything that made it like that?

But maybe you know best how to handle it?

well what kind of awkward is it? like awkward because feels like you two don't have anything to talk about or awkward because don't know what topic to talk because afraid of knowing another truth?

i also same with you, the type to overthink something and having negative thoughts. aaaand because of that i always make sure to ask what's wrong, what's the problem, let's talk about it right now. so the negative thoughts will go away if we already talk about it. and if you ask ur friend what's wrong and they told you, you have to make sure to take it good, like being open-minded.
in my opinion, many people didn't tell someone/their partner what's wrong because sometimes it may lead to worse situation because the person isn't being open-minded.

maybe this can help? lol

I don't understand why you would talk about "seeing other people" in the first place? It may just be that it was hard to explain or whatever, but I assume that everybody who is in a relationship knows, understands and accepts that their partner has friends and meet new people, maybe not on a daily basis but still quite regularly. Which of course should lead this discussion to "are you seeing other people romantically?" - and why would you have that discussion if you weren't thinking of going polyamorous.

So I see this going three ways; your partner thinks you want another partner or your partner is poly and wants more and doesn't know how to bring it up or your partner is insecure about themselves in your relationship and doesn't feel adequate.

Considering that your partner said 'talking can lead to something else' I believe it's either the first or the last.

Awkwardness usually means there's more to talk about, so before you do anything and runs away you should ask them what it's all about. Because you want them to tell you what's wrong - so shouldn't you at least make it possible for them to do so?
This seems like there is definitely something that's harder to talk about so if you run away without even giving them the chance, the conversation-opener and maybe helps ease them into the core of the problem - did you ever give them the chance to tell you what's wrong?

I think communication (as always) is the best way to go and stop analyzing too much on it. Talk first, give them the opportunity to say what bothers them and make them feel secure enough to actually do this.

That is honestly the best way to react. (And still - I don't get why you even had this conversation to begin with if there wasn't an ulterior move behind it.)