Your instinct about not rounding early is good. Before I get to that, I wanted to address one thing in another answer:
"> I learned that the absolute and relative error have only 1 significant figure
That really isn't true. In your example you showed 5.34532g ± 0.001428g which could result from some kind of experiment. The point being that the digits 1428 could all be significant."
Theoretically possible, but pretty unlikely. To the best of my knowledge, even the National Institute of Standard and Technology never give more than 2 digits of uncertainty because the number of measurements required to give a 3rd digit any meaning is unreasonably large.
The 1 significant figure in error "rule" is pretty good. There are two exceptions. 1) If the uncertainty is very well known (because it was calculated based on a large number of measurements), two sig figs can often be justified. 2) If the first digit in the uncertainty is a 1, two sig figs are usually justified. Think of it this way: Rounding ±0.16 to ±0.2 changes the uncertainty by 25%, while rounding ±0.86 to ±0.9 changes the uncertainty by only 5%. In many cases, the uncertainty is known to better than 25%. It's rarer to know it to 5%.
So, one or two sig figs are justified in the answer, but what does that mean for rounding? A good guideline is to keep at least one more digit than is significant until the last step. There's nothing really wrong with keeping even more, they just are unlikely to make a difference in the final (rounded) result.