I agree with above post, you can by RaW cast gfb off shadowblade but can’t twin gfb as it can hit multiple targets, it should be noted that boomingblade CAN be twinned, as it is a single target spell.
It should also be noted you should ask your DM if you can take both greenflameblade/boomingblade AND shadowblade, as shadowblade is from xanthars guide, while the other two are from the sword coast adventures, it would violate the standard phb+1 in AL, and some DMs prefer to use this ruleset as well so as to prevent gamebreaking builds, and while it’s fine at base level upcasting shadowblade while applying the cantrip damage, can get pretty nutty at higher levels for sorcerer, assuming you maintain concentration, and upcast, at maximum level
Example: with a 7th tier slot at lvl 18 the combo can deal 5d8 psychic+3d8 fire and 3d8fire+mod to a secondary adjacent target per round, if you quicken AND twin boomingblade after that’s an additional 5d8 psychic+3d8thunder to two targets in range plus an additional 4d8 thunder to each if they move before your next turn.
That’s a possible total of 35d8+3x str/dex mod+1xCha mod damage between two adjacent targets for a single 7th tier slot, every turn after you summon your blade, where you lose out on the gfb strike. then the combo only costs 3 sorcery points for every turn after, which you can do for 5 more turns. All For Sorcery points and a single 7th tier slot if you maintain concentration.
And thats WITHOUT taking a 2 level dip in paladin for divine smite.
Assuming max upcast At lower levels and accounting for cantrip scaling it’s still up to
29 d8+3x str/dex mod+1xCha mod at lvl 14 with a 7th tier slot.
26d8+3x str/dex mod+1xCha mod at lvl 11 with a 5th tier slot.
20d8+3x str/dex mod+1xCha mod at lvl 9 with a 5th tier slot.
17d8+3x str/dex mod+1xCha mod at lvl 5 with a 3rd tier slot.
8d8+3x str/dex mod+1xCha mod at lvl 3 with a 2nd tier slot.
Granted you can only use the combo with 3 sorcery points, but the higher lvl you get the longer you can potentially sustain this ridiculous output, and you can always swap out the gfb spell for something like fireball or disintegrate, or just twin boominblade for 1 sorcery point a turn for around 70% of the previously shown damage output.