He regularly went to the grocer, and he had his own money. The issue would have been, as others have alluded to, that any letter sent to a northern state with an unfamiliar address would have been 'noticed' and it is probable that would have been reported back to his 'owners'.
It night have worked, but the result of a failure would have likely been fatal.
What I don't get is why in 12 years Solomon never encountered a Quaker or any other religious figure of a persuasion known to be abolitionist. I accept they would have been rare birds in the South, but it seems odd to me that the first 'friendly' assistance he finds is a guy from Canada, of unknown disposition, (until said Canadian has an argument with the master).
If it were me, I would have left a note in the first Quaker meeting house I came across. Or similar.
In USA, it was "ok" to have/own slave, and backed by the law; just like Islamophobia is ok now. The fact that Solomon found the 1 guy, who was against slavery, is a Canadian clearly reflect that sad reality.