In other words, did Paulie and Carlo really do the things for which they were killed.
Absolutely, they were guilty of betraying the Corleone Family but for different reasons.
Note: Very little of this is covered in the movie. Unfortunately, in terms of the movie these are minor characters (Paulie more so that Carlo) and trying to cram the whole novel into a movie would have made it significantly longer...something in the order of 200-300% longer.
Hence, my references/inferences will be from the source novel by Mario Puzo.
I'll start with Paulie because he's the simplest to cover.
He's relatively low-level in the Corleone organisation but still trusted after 'making his bones'...
Michael's thoughts on Paulie...
Paulie as yet had not become rich. He was well thought of, his rise in the organization was certain but he would have to put in his time like everybody else. Also he would have wilder dreams of power, as the young always do.
I've covered how Paulie was 'convicted' in an earlier Q&A but his motivation is not covered there.
For Paulie Gatto it was a case of greed. The novel tells us that..
..after he had “made his bones” he had received a good living from the Family, a percentage of an East Side “book” and a union payroll slot. Clemenza had not been unaware that Paulie Gatto supplemented his income with free-lance stickups, strictly against the Family rules..
Clemenza had several times spoken to the Don about better rewards for the all-important button man who was first in the front line when trouble arose, but the Don had put him off. If Paulie had been making more money, he might have been able to resist the blandishments of the wily Turk, Sollozzo.
When they are driving as part of the ruse to take Paulie to his fate...Clemenza thinks...
As he had expected, Gatto’s eyes became greedily interested. Paulie had swallowed the bait and because he was thinking how much the information was worth to Sollozzo, he was forgetting to think about whether he was in danger.
For Carlo his motivations are more personal.
He's also greedy but, for him, it's a matter of being, in his eyes under-valued and unappreciated by the Corleone Family.
Carlo Rizzi smiled. It was only the beginning. He had, after all, married into a royal family. They would have to take care of him.
Of course, the Family's perception was quite different..
And Carlo Rizzi was turning out to be a real loser. He had been fixed up with a nice little business and was running it into the ground. He was also drinking, whoring around, gambling and beating his wife up occasionally.
Carlo Rizzi was a punk sore at the world. Once married into the Corleone Family, he’d been shunted aside with a small bookmaker’s business on the Upper East Side of Manhattan. He’d counted on one of the houses in the mall on Long Beach, he knew the Don could move retainer families out when he pleased and he had been sure it would happen and he would be on the inside of everything. But the Don wasn’t treating him right. The “Great Don,” he thought with scorn. An old Moustache Pete who’d been caught out on the street by gunmen like any dumb small-time hood. He hoped the old bastard croaked. Sonny had been his friend once and if Sonny became the head of the Family maybe he’d get a break, get on the inside.
But Carlo was also foolish in that, although the Family tolerated his infidelities and, to a certain extent his treatment of his wife, he went too far.
In fact he bragged..[snip].. about how he bounced his wife around when she got snotty and saw their looks of respect that he had the guts to manhandle the daughter of the great Don Corleone.
..but he understimated his 'friend' Sonny's reaction.
But Rizzi would not have felt so safe if he had known that when Sonny Corleone learned of the beatings he had flown into a murderous rage and had been restrained only by the sternest and most imperious command of the Don himself, a command that even Sonny dared not disobey. Which was why Sonny avoided Rizzi, not trusting himself to control his temper.
...now Carlo has been humiliated by Sonny which adds additional motivation.
However, the "betrayal" by Carlo is a 'twist' in both the novel and the movie but the novel tells us that the behaviour of Carlo was instrumental in the assassination of Sonny.
But the enemy was making its plans. They too had analyzed the situation and had come to the conclusion that the only way to stave off complete defeat was to kill Sonny Corleone.
This paragraph is immediately followed by the call from an anonymous woman to Connie (Carlo's wife) which sets her off and leads the row between them and, eventually, to Sonny being killed.
It’s hard to see why either Paulie or Carlo would expect to profit from these plots, even if they expected to get away with it, as seems very unlikely under the circumstances.
It's likewise hard to see why they would not try to disappear from Corleoneville.
They have no reason to suspect that they were themselves suspected. For Paulie, he underestimates the intelligence network of the Corleones and Carlo is apparently incidental to (although the proximate cause of) Sonny's death.
Leaving would have been a definite sign of guilt and they would have been hunted down by the nation-wide reach of the Corleones.
Keeping one's head down and hoping that no-one finds out would, I suspect, the most logical course of action.