Also you might not have noticed yet, but you have an additional Storage Box, which works exactly as it sounds. Its a secondary inventory with 200 slots which can be further upgraded in the same way as you basic inventory.
Its a great way to make sure you have plenty of everything needed for evolutions / blooming. Whenever you get a evolution Dragon, a Bloom Chicken, petals, etc just throw them into the storage box. Not only you`ll save space in the normal inventory but also will make sure you have plenty of everything to go around.

Its also a good way to keep them ampys around when you are still waiting for that one 6* or your 5* waifu whom you wanna max out.
A good thing is to take a loooong look at the exchange center tabs. If you are solid on cash, you can for example buy 4* girls and the blooming petals on regular basis.