Apparently the 3rd of the VI needs to be doubled but in this progression I have written the motion from V to VI with the normal doubling and it sounds great to me. I don't understand why they make these rules. Can someone please tell me what part writing error I have committed?
You have an augmented second between F♯ and E♭, and so that's the part-writing error that you have present. Moving this F♯ up to G will double the third of that VI chord, as you mentioned.
I always teach my students the following in a deceptive/interrupted resolution: the bass and leading tone go up, but everything else goes down. This is the only way to prevent parallel fifths, octaves, and improper augmented seconds.
And frankly, I also do this in major keys, even though it's not necessary. (There scale-degrees 7 and 6 are only a major second apart, not an augmented second, so the leading tone can go down to 6.)
I think you're pretty much stuck with a unison G in the top two parts here. The leading note F♯ badly wants to rise. In a perfect cadence there's the 'fall to the 5th' exemption, in favour of a full final chord, but that's not an option in a V - VI progression. And while I LOVE the astringent quality of an E♭ - F♯ augmented 2nd in a melodic line (and it's not THAT hard to sing) it isn't textbook, and has no melodic virtue here anyway. Sing your alto line to see what I mean.
Doubling the 3rd of a minor chord is acceptable. Particularly when, as here, it's the tonic note of the key.
How about making the first chord a D7, with C in the tenor rather than D? Or a pair of 8ths, D and C? That could sound more interesting.