I'd say not terribly. While there was no written constitution, the Romans had quite a thing for tradition and didn't change things lightly. The downfall of the Republic stems from the fact that ultimately it was a system of government designed to run a city-state now being used to run an empire filled with ambitious and obstinate men. The defining moment that started the collapse was Sulla's march on Rome - even if they had a written constitution it wouldn't have stopped an ambitious general with an army at his back.

It's not as though the Romans didn't have written laws. The reason we say that there was no written constitution is that there was no document that laid down what was legally permissible in terms of the passage and structure of bills, or that laid down explicitly exactly how and within what boundaries the state was to act. The United States Constitution isn't really a law, although it has the force of law, it's more like a framework for the formulation of laws and powers for the future. For the Romans the framework was the laws, the res publica. Roman laws generally didn't get superseded, they were just replaced or updated, whereupon the older laws would simply be ignored. For example, many of the laws on the Twelve Tables were still technically legal in Caesar's lifetime. While many of them had been contradicted, like the law against marriage between plebeians and patricians, they had never been thrown out. In the modern US legal system we discard laws or parts of the Constitution that have been amended or superseded--the Romans just ignored them. Often laws remained from as early as the Twelve Tables that had long been rendered obsolete. Famously Caesar's opponents stalled senatorial debate by raising a flag on the Janiculum, citing an absurdly old legal practice from Rome's days as a city state, that stated that all senatorial debate must end when a flag was raised on the hill, which was the signal that an enemy army was in sight. Labienus tried Rabirius under the long-outdated charge of perduellio, which is in the Twelve Tables but had never actually been replaced, even though nobody had used it in centuries.

The instability in the unwritten constitution is that, while in general things really didn't change much, new laws could be added that changed everything and old laws sort of ended up in a big jumble with them. You couldn't just go around abolishing magistracies and stuff willy-nilly--the traditions of the state needed to kept intact, for ritual reasons along with others (the mos maiorum is inviolate, and breaking it is a ritual offense), and even in the Principate we find that traditional magistracies and legal bodies are still intact, it's just that there's somebody watching over them. But you could have people like Sulla, who radically altered the way that the magistracies of the state worked even by just doing little things like increasing the number of magistracies, the number of senators, and forbidding equites from serving as jurors. In the same vein, Pompey and Crassus during their consulship in 70 totally tore apart the Sullan Constitution (with pretty much everybody's blessing, it was hated by everyone), leaving only a few things like promagisterial commands in the provinces--the obvious problems when there's no fundamental, foundational framework within which the laws have to operate becomes apparent when we see stuff like this.

Still, it's not as though the Roman state was rocked with laws changing left and right throughout the centuries that the Republican state existed. Laws didn't really change that much until the 1st Century, when the combination of social stresses, economic stresses, ambitious individuals, and equally ambitious (and very much politically motivated and active) segments within society began bumping into each other. The only analogous situation earlier was the Conflict of the Orders, which was resolved relatively well--in some ways, as Badian pointed out in his review of Meier's book on Caesar, the bizarreness of the Roman state allowed to be actually more successful at reimagining itself and adapting than many other states since. This coming from Badian, a devout supporter of Syme's, should speak quite highly towards the idea of the flexibility that this ideally allowed the Roman state--the downfall of the Republic was not due solely to its lack of a written foundation, but due to a number of combining factors, some of which are wrapped up in the legal system but others of which aren't really