Okay - so if I understand your question correctly you are attempting to use your chart to transpose a song from one key to another. But the problem arises whenever the song you are transposing contains chords that are outside the root key - so the chart does not line up.
So here is what you do. Use your chart to find the chords based on the root note indicated in your chart by reading from the appropriate row as you indicated. When you get to a chord based on a note that is outside the key (non-diatonic), you need to determine the intervallic relationship between the preceding chords root and the chromatic (outside the key) chords root. In other words, how many semitones (half steps) separate the two root notes.
To easily figure out the answer - use the chart below and simply count the number of steps from one note to the other. Then count the same number of semitones from the note in the new key to find the next root note for the corresponding chord in the new key.
So using this method with your example - let's say you are transposing from your song in the key of E to the key of F. Your first chord is a D# Seventh Augmented Ninth Diminished Fifth. The root note of this chord is D# and is in the key of E so to transpose to the key of F you look at your chart and see that D# is 7th scale degree of the key of E so you go down to the 7th scale degree of your new key F and see that you will need a E Seventh Augmented Ninth Diminished Fifth. Next is the G# chord. From your chart we see that G# is the 3rd scale degree of E so for the key of F we will look at the 3rd scale degree and find an A chord.
But as you have discovered - when we get to the next chord (F) which is outside of the key of E - we can't find the F in the line for notes for the key of E so what do we do? This is where the new chart comes into play.
We look at the chart above and find the G# and count the number of half steps (semitones) between G# and F and we see that there are 9 semitones if we go higher (you must loop back to the beginning when you get to the B and go B to C - just like on a piano keyboard). If you take the new key of F and start with the note that preceeded the chromatic chord root which was A (see above) - and count 9 semitones from A on the chart above we land on Gb (also known as F#). So the chord we need in our transposition from E to F is Gb. Alternatively we can go in descending steps down from the G# to the F and see that there are 3 half steps or semitones between G# and F. So we can start on A and count 3 half steps to the left and again we land on Gb/F# - so it works in either direction.
EVEN EASIER - You will also note that the out of key chord root in our example amounts to a "flatted" second degree of the scale. So instead of the F# we have F which is the 2nd degree flatted. If we go to the new key of F and look at the 2nd degree of the F scale on your chart we find G. If we flat the G by one semitone we have Gb.
You can use the chart above to discover this as well. For the example in your question - simply take the 2nd scale degree of whatever key you want to transpose to and count one half step/semitone down and that gives you the equivalent flatted second degree for the new key. Use the corresponding root for the equivalent chord (be it major, minor, 7th, diminished, etc.) and you have the appropriate chord in your new key. This will work for any key you choose to transpose to.