I think it helps to visualize what happens if you move all your outfield checkers forward by multiples of 6 into your inner board. For checkers in your inner board, keep only the checker on the lowest occupied point. This gives you:
| White | 0 |
|---|---|
| |
| Blue | 19 |
Saving the gammon is similar to trying to bear off all your checkers from this position, with two differences: (1) there really should be a stack of 15 checkers on the 6pt, representing all the crossovers you need, and (2) the legal moves aren't quite the same. Despite that, I think the exercise is helpful because it highlights the gaps in your position that are likely to cause "misses" later on.
Consider a move like 13/10(4). This will in effect take four checkers off your 6pt stack at the cost of moving the four checkers on your 1pt back to the 4pt. This eliminates one gap at the cost of creating a new one. Intuitively, I think it ought to be better to play 13/10(2) and look for two 3's elsewhere, because 13/10(2) smooths your distribution from 4 checkers on the 1pt to 2 checkers on the 4pt and 2 checkers on the 1pt. The remaining two 3's can be used to smooth your stack on the 2pt. One of them needs to be played 8/5. The other one could be played 8/5 or 20/17. I'd guess that these two plays are pretty close. The main difference is that 8/5 brings another checker into the homeboard but wastes a pip.
In summary, I think the two best plays are probably 20/17 13/10(2) 8/5 and 13/10(2) 8/5(2). Not sure which one is better.