Should I update to achieve more accurate transpositions?

As far a double or single only is concerned. If you select single only, then transposition would change any doubles to the corresponding natural (Bbb = A for example).

Version 0.9.6 ?!?!?!?!?!?!? You've had that a while. Since I didn't use anything before 1.3 and don't do lead sheets, I'm not sure what the benefits of upgrading would be. It's possible that version is sufficient with few improvements, but I'm sure someone who knows will have input.

Double sharps (and flats) are inevitable in the some Keys (especially minor).
Example: G# minor harmonic. (Relevant to B Major)
G# A# B C# D# E Fx G# (Fx = F##)
sansible (leading-tone, 7th degree of G# harmonic minor scale) is f##.
If you want to avoid this, the software will write G instead of Fx. (Theoretically not the right notation.)

Or when you are in E-minor, in melodic line, you have "d, db, c".
When transposing to the Eb minor, the result will be "db dbb cb".
If you want to avoid this, the software writes "db c cb". (dbb <=>c is enharmonic) // ( again: Theoretically not the right notation.)

This is what is mentioned in the transposition section.

You should definitely upgrade... MuseScore 0.9.6.x has been released in 2010, 7 years ago... A lot of work and effort has been put into the different versions, it should be night and day, even if you are only doing leadsheets.

From my perspective...
I still use lead sheets printed back when 0.9.6 was current. (Plastic sheet protectors used in a 3 ring binder.)

Here's a scan of an excerpt from one of my 0.9.6 scores:

If you look closely, you'll notice that the flat signs in the chord names are not all the same. Each substitute chord (with parentheses) does not display a 'legitimate' flat sign, which means that it will not change should the entire score be transposed to a different key. (For those substitute chords, I used a Windows font called Comic Sans MS to make the 'unrecognized' chords look similar to the 'actual' MuseScore chordname font.)

Also, back in the 0.9.6 days, there were a number of chord .xml files, each of which recognized a variant spelling of a chord name. A user had to select among 'styles' of chord spellings and enter chordnames accurately -- many times with his fingers crossed: if typing a lower case letter produced a capital letter; and typing a 'b' produced a flat sign; and typing a '#' produced a sharp sign, all was well. (Sometimes the chord file had to be opened in MuseScore just to check how the chord should be properly spelled in order to be recognized.)

Since those days, I have migrated that very score through subsequent versions of MuseScore.
Today, that same snippet looks like this:

(BTW: This picture was taken with the MuseScore image capture feature.)

See? The flat signs are all recognized. The font is truly the same for all the chords; and everything transposes correctly, if needed. This happened 'automagically' -- actually, here's where Marc takes a bow ;-)
So... the chord parser alone was one giant leap forward.
Also, if your lead sheets are akin to 'songbook-style' sheets, nowadays there is a fretboard diagram feature.

Whew! ...This is only a modest example of what has improved since 0.9.6 !!!

P.S. For you, though, there might be a learning curve. For example, the spacebar is no longer used for entering rests. (It starts/stops playback - similar to contemporary A/V software.)
Also, there's this thing called THE INSPECTOR... yikes!

Regards.