Say I have AJ in a 1/3 game and called a raise and it winds up HU. The flop comes Kxx, Qxx, or xxx and the raiser cbets half pot. In a tournament, I might bluff raise or float if I thought he was just cbetting. Here, the raising ranges are stronger and it is harder to get people to fold. Should I make a play or just give up?

So two things.

1) at most 1/3 games, people raise 5xBB+ so 3 bet or fold is a much better strategy than doing any sort of cold calling. When i do, usually its to close the action HU from the BB

2) when I do call a PFR and it gets HU (usually this is at 2/5), against fishy players I tend to float a very wide range, mayne even 90%+ of my range vs certain Vs. You should look at gto’s calling range in various flop spots and see how wide they call. I dont think much of anyone here is finding calls or raises with hands like 9c7c on a Kd8s3c board. But exploitatively against a group of Vs who over cbet and under turn harrel, floating IP is even more profitable with even weaker hands

As a general principle you should make plays less often in cash games then in tournaments because your opponents will be making fewer plays. The tournament time limit forces players to make more bluffs while at the same time making it riskier to waste any of your stack.
This doesn't mean no floating but there should be less. How much is very villain dependent and varies more then tournament play.
As for the bluff raises, in cash because there is less flop bluffing and no time limit there is less reason to make counter-plays on the flop. If your opponent is bluffing you might as well let them lose more money bluffing before countering them. The bluff raising usually happens on the turn.

A lot of low stakes players c-bet too frequently, and will often be one and done, so it pays to float somewhat wide when we're IP.

We can float wider when they c-bet smaller, and with hands that have more potential to catch up on later streets.

So I might float with AJs on a K high or Q high board if we have the BDFD and if V bets 1/3 pot. I'll float with AJs if we have the FDFD if V bets 1/2 pot. If we have AJo, I'm mostly folding to any bet size, but might float a 1/3 pot bet if the ace in our hand blocks the NFD on the front door side, with plans to turn my hand into a bluff on a flush runout.

Hope that helps.

Lets look at GTO. CO vs BB.

On a Ks6d5d board, the CO only cbets about 55% of the time. Almost nobody cbets that little. It then calls about 55% of the time including all gutshots, AJs ATs with BDFD, any Ad, etc. due to the MUCH wider cbet of V here though, you should be folding less often.

The turn is where it gets really interesting. Lets say the fund is an offsuit T. CO should be barreling 60% of the time, a lot of that as an overbet. This includes betting hands like J7s Q7s with absolutely no flush draws, its effectively turning hands like Q6 and Q5 into bluffs at some frequencies etc. NOBODY does this. You are going to get to see the river WAY more than you have any business seeing the river, which, beyond bluff opportunities, means that a total junk hand like J9 might just hit a runner runner straight or just a random pair to win.

Illuminating

Not nearly enough info in the scenario you described. How deep? What positions? Player profiles? Generally no this is not a thing, unless you have specific reads that villain is over cbetting and then folding a lot when raised. But depending on the board, you may have better bluff candidates and AJ may be strong enough to float depending on how tight the ranges are and the player you are up against. IE against a competent player defending the big blind vs button, AJ is probably good enough to defend on a lot of Qxx boards. Not as much UTG vs BB. Also, when floating, we want to be against the type of player that will check back turns / give up on later streets. Against someone who will fire multiple barrels frequently with hands with a lot of showdown value that are strong enough to check, we should just fold earlier streets. IE a player that is going to bet 66 on Q82 and barrel on a turn 5, we should just be folding the flop.

But vs some players who really over cbet you can consider bluff raising any two. Usually I would attack a 1/3 pot bet more than 1/2 though, and bigger bets we should raise less often.