There is something kind of Paleo Conservative about the idea that we are destined to be in direct competetion for resources.
This has it kind of backwards. Competition is what gives people a reason to make progress. Getting rid of competition is a way to make it practical to keep a society static. Can't get much more paleoconservative than that.
There will be, and should be new ways of going about things.
Well, a species with a hive mind wouldn't compete. Is that what you think we should go for?
Short of that, if we don't compete, then that means there's an overriding goal we all cooperate in pursuit of -- some common goal we all subordinate our individual goals to. Is there some goal you think people could all agree to make their top priority?
Short of that, if we all cooperate in pursuit of some goal and subordinate our individual goals to it even though some of us don't think the collective goal ought to be our top priority, who should get to decide which people get to make their goal the common collective goal, and which people have to cooperate in pursuing a goal that isn't their own?
I.e., have you thought this through? Why do you think a new way of going about things without competition is something there should be? It sounds to me like it would be a dystopian nightmare.