To answer your second question first: yes, Charles' Death Warrant still exists, and is held by the Parliamentary Archives in the United Kingdom.

As far as the risk the signatories of the death warrant faced, at the time they faced very little. It wasn't as if he was still at large, at the head of a Royalist army, when they decreed his death. By the time the warrant was signed, Charles had been in Parliamentary custody for months, and the warrant was signed at the end of his trial for treason. Due to the unprecedented nature of the trial, a special commission of 135 men had been established to judge the king (despite Charles refusing to accept the validity of the trial in the first place), and by the end of the proceedings only 59 of the commissioners signed the warrant. Some had refused to participate at all, while others were involved with the trial but could not countenance the death of the king. But Charles was duly executed by beheading on 30th January 1649 at St James' Palace, and for over a decade the signatories of the warrant, the Regicides, faced no consequences specific to their involvement.

And then... Charles II, father to a murdered son, husband to a murdered wife, son to a executed father, was restored to the throne in 1660, and their signatures became an enormous liability to their general wellbeing. By 1660, many of the Regicides had died, including many of the instigators of the trial, such as Oliver Cromwell and John Bradshaw, but many more still lived. Some of these men fled the kingdom on the return of the monarchy, usually either to English colonies in America or to the Netherlands or Switzerland. Charles neither forgot nor forgave, and royal agents relentlessly searched for those who had fled. Relatively few escaped the king's wrath.

Those who stayed in England, or who were caught and returned, faced execution, or had their sentence commuted to life imprisonment. Those executed faced the traitors death: to be hanged, drawn, and quartered. Four of the Regicides who died before the Restoration (Oliver Cromwell, John Bradshaw, Thomas Pride, and Henry Ireton) were posthumously attainted and sentenced to the same fate. Cromwell, Bradshaw, and Ireton's remains were exhumed and subsequently dragged to the gallows, hanged in a noose, and then decapitated with their heads being displayed on spikes. Pride's body was not treated this way, possibly because it was too decomposed.

So the Regicides were taking a huge risk that, should the Stuarts return to power, Charles' death warrant would become their own. But in the immediate aftermath, it wasn't clear that the Stuarts would return. It took a decade, and some quite unlikely events, to bring about the Restoration.