Yes, I believed him, for reasons many of which have already been stated in this thread. In addition, Cullen (who, for the record, is incensed at Rainier's actions) notes that he decided to confess just when he had effectively shaken off his past--he had everything to lose by turning himself in and fully expected, and wished, to be executed. He was clearly trying to come clean, and therefore it would make no sense for him to deny killing the Warden Constable if he had done so. And as others have already pointed out, the real Blackwall's death was not convenient for him at all. Living under a false identity is very risky, even if you're careful.
Also, as Nixou has pointed out, Cole would know if Blackwall were lying. If you travel around with both of them in your party after Revelations, they have a number of banters about Blackwall's past. Some of these may depend on Cole being more spirit, as one of the ones I heard started out with Blackwall questioning Cole's decision to forgive the Templar who had killed the original, human Cole. He objected to Cole's making the Templar forget what he'd done, and Cole said he understood that Blackwall was afraid that if he forgot his crimes, he might become that way again. There's also a banter in which Blackwall asks Cole if he's angry at him now that he knows the truth, to which Cole replies that he always knew. When Blackwall asks why Cole didn't tell on him, Cole says he didn't because Blackwall wanted to change. Cole states unequivocally, "You are Blackwall. You killed Rainier," and goes on to say that he knows that if Blackwall could go back, he would stand between his younger self and the carriage to save Callier and his family. What more proof do you need?
I also disagree with your assertion that Rainier met Blackwall "at the exact time when murder and dishonesty were what he was about." Some time passed between the Callier massacre and when he met Blackwall, and if you ask him about how they met, Rainier says he was basically drinking himself into a stupor. I get the impression that was his habit at the time. It's clear (again from a banter with Cole) that he was horrified to realize Callier's children were with him, and it sounds like in the moment he just froze, unable to process the situation he'd put himself and his men in. By all indications, he was consumed with guilt right after the massacre but yes, he was also a coward at the time, so he hid. He wandered the land, trying to drown his guilt in alcohol because he didn't know what else to do. Warden Constable Blackwall offered him a sense of direction he had previously lacked. And if you doubt this, look at his actions at the time you recruit him and his behavior while he's with the Inquisition. He shows an obvious devotion to helping and protecting people who can't help or protect themselves, to a degree that is not required merely in order to masquerade as a Grey Warden. Grey Wardens are not required to protect farmers from bandits or to approve of the Inquisitor doing similar things.
His compunction is obvious. You can certainly consider it inadequate, and if you don't feel that his efforts at atonement expiate his crimes, then you are free to leave him in prison and let him hang. But there is really no room to doubt his sincerity when he finally confesses.