Shyv needs a semi-rework. Not total. Just some changes to make her kit more interesting and viable compared to other jungle champs. Her dragon form is very underwhelming, and as you say, actually using it gives you very little other than a sudden scare when she jumps a wall to drop-in on the other team's back-line carries.
Shyvana's Q and W both scale off of AD. W does magic damage while Q does physical damage. But her E and R scale off of AP. Her E, when it hits (skillshot), gives 2% bonus magic damage to her AAs based on the hit champs's max hp. Essentially, no form of itemization is efficient on Shyvana except tank because her scaling is all over the place and she has a passive armor/MR buff that is doubled while in dragon form. Split scaling is bad, m'kay?
In short, Shyvana lacks focus and a clearly defined identity.
The other consideration is to make her dragon form more distinguished from her human form. Currently, her dragon abilities are very underwhelming and do nothing different other than a semi-AoE. The Flame Breath isn't even a true cone! It's 5 projectiles fired at angles in front of her. Her Q has no visible effect at all! They need to be more identifiable with being a dragon, rather than just slightly different from human form. This does not necessarily mean more powerful.
First I would change Dragon's Decent damage scaling to do physical damage based on her hp. She's a dragon jumping on top of someone. Why does it do AP scaling damage? Makes no sense. %HP would synergize much better when she builds tanky.
Next, give her Q a passive that stacks hp based on monster and champ kills. If she last hits with it, she gets a stack. Give some HP for each stack.
Second, she needs something to distinguish her while she is in dragon form. She is a dragon! Why she no fly? Make her dragon-form W a gap-closer/escape such that she leaps and glides a distance similar to Corki's Valkyrie. Leave the AoE flames and speed boost as they are. If the speed boost got removed I wouldn't cry. The leap would be compensation enough I think. But honestly, DRAGON! She should be bigger and faster than those puny humans puttering around under her feet.
These changes would reward Shyvana players for building along her best stats, as well as allow meaningful and focused itemization, rather than having some of this and some of that. It would also distinguish her dragon form from her human form in meaningful and clearly appropriate ways without being OP. The gap closer in dragon form would make her very sticky in ganks, something she currently lacks.
It would not allow her to gank any easier pre-6. But that is not necessarily a bad thing. It simply means that that part of her identity stays the same. She will continue to be a tanky farming jungler that leaps in and disrupts the enemy team with subpar AoE melee dps. But her ganks will have some teeth (pun intended), her dragon abilities will actually have value, and she will have an escape/gap-closer.
Power-wise, I think this would put her just below Jax, but focused more on AoE than on soloing.