Dear Bernhard,
Could you specify the Scarlatti sonatas which address the matter, double thirds and/or any double notes?
Do you have some other intermediate repertoire suggestions to work out double-notes (including at the LH)?
Thanks by advance!

It is almost easier to answer the reverse question: which sonata does not.
Here are just a few:
K1 - thirds on the left hand.
k18 - double notes (thirds included on right and left hand)
k 24 - thirds on right and left hand.
k 46 - thirds on left hand (mostly)
k 56 - thirds on right hand
k 57 - double notes on left hand
k82 - thirds on right hand
k84 - double notes everywhere (thirds and sixths)
k86 - double notes (mostly thirds on right hand)
k87 - double notes (mostly sixths)
k92 - lots of double notes (all sorts)
k96 - chords, octaves, thirds, sixths.
k100 - thirds on right hand
k104 - thirds, octaves, chords
k105 - as K 104
K 107 - thirds on both hands
k108 - thirds and octaves on both hands
k115 - double notes on left hand
k 120 - lots of double notes (thirds and sixths) on right hand
k127 - thirds, sixths and octaves on left hand
K 133 - octaves (and a few thirds)
K 134 - thirds on right hand
K 139 - double notes of all kinds (including seconds)
K 140 - lots of double notes on the right hand
K 141 - double notes and chords on left hand
K 143 - double notes on the right hand (thirds, sixths and octaves)
K 147 - lots of double notes on the right hand
K 145 - sixths, sevenths and octaves on the left hand
K 162 - all sorts of double notes on left hand
K 164 - thirds and double notes on left hand
K 166 - double notes on the left hand
K 169 - as K 166
K 170 - as K 166
K 172 - double notes on the right hand
K 201 - double notes on right hand
K 202 - double notes and chords on left hand
K 207 - thirds on right hand
K 236 - lots of double notes on both hands
K 238 - as K 236
K 245 - thirds mostly on right hand
K 247 - thirds on left hand
k 256 - all sorts of double notes on both hands
k 280 - double notes on left hand
K 286 - double notes on left hand
K 288 - thirds on right hand
K 294 - thirds on left hand
K 299 - lots of double notes in both hands
K 301 - double notes on left hand
K 364 - octaves on left hand
K 370 - thirds on both hands
K 372 - thirds and sixths on right hand (a few on left)
K 392 - fifths, sixths and sevenths on left hand
K 394 - double notes on left hand
K 402 - double notes on both hands
K 407 - thirds on left hand
K 436 - double notes on both hands
K 444 - As k 436
k 450 - lots of double notes on both hands
K 481 - double notes on left hand
K 487 - left hand octaves
K 491 - double notes on right hand (lots of thirds)
K 492 - thirds on right hand, octaves on left hand
K 520 - thirds on right hand
Enjoy

Best wishes,
Bernhard