I’d only correct the original data if I had sufficient information to justify a change. I wouldn’t assume my GPS trace is any better than the original source. I’d only change the original if there was enough GPS traces to justify the change (from different users) or there was something obviously wrong with the original data - for example when you stand in the road you can see it is straight but the the data suggests a zig zagging road for example.
As to problems connecting your traces to the rest of the data. I’d probably favour connecting it, even if the connection wasn’t perfect. I tend to view the connection of more value than having a road in slightly the wrong place. In this case if the necessary correction was more than minor then I’d probably add a FIXME tag with an explanation so that somebody can see the place needs investigating.