Your AC is bad. Disadvantage is your best friend
You seem to have a very low ac which seems to at least be saved by the fact that Draconic Sorcerer's have a built in Mage armor but you don't have access to use shields to pump it up higher. If you really want to do a melee sorcerer you are going to need spells that make it hard to hit you. Or at least make it so enemies don't want to hit you.
Spell Option: Blink
Blink is a pretty good spell to think about cause of the effect of possibly transporting you into the ethereal plane until the start of your next turn. But the downside is it's a literal roll of the d20 if you get this effect again at the end of your turn. But upside is you don't have to worry about concentration.
Spell Option: Mirror Image
Mirror Image creates three illusory duplicates of yourself with an AC of 10+ your dex and when enemy makes an attack at you you roll a d20 to see if the creature attacks you or your duplicates.
Downside they still have a chance to hit you if you have bad luck. And two with your dex and the level you are playing it just means you have three body bags that disappear with one attack. However this is also not concentration so you can stack it with haste and blink if you choose.
Spell Option: Polymorph
Polymorph is a way to use your concentration slot which you will be able to accomplish being a pseudo-tank by choosing to become a very Hearty beast with a high con mod which can allow you to keep concentration and tank health in your morph form like a druid and their wild shape ability.
Like you can become a huge T-rex and have its multi-attack if you are trying to be up close and personal damage dealer other than touch spells. And you also have the benefit of not being stunned which losing concentration with Haste does.
Spell Option: Blindness
Blindness would be a good spell to throw at enemies as you are able to make them attack with disadvantage with no cost to concentration.
But it's a Con Save and at your level you have a 15 spell save dc which means you are not going to be as effective as think it is as most higher end creatures tend to have either blindsense which negates blindness or relatively high Con saving throws.
Resistances to elements
Of course elemental damage is the more explosive and heavy hitting there is in DnD typically (ie Fireball, Fire Dragon's breath, Magic Missile if built out like crazy). Most enemies and spellcasters has a pension for Fire Damage so go for a Fire Ancestry dragon and to help cover other elemental damage types. Absorb Elements is great as its basically an elemental equivalent to the shield spell you also have the benefit of adding a d6 of that damage type to your next attack.
Haste Is Honestly Not Optimal On You
Haste is an amazing spell but something to keep in mind it's also a double edged sword for when you drop concentration the target in question is stunned. It is your character but you should understand the risk on casting it yourself even if you do have an effective +10 to concentration checks which can happen alot when you are in melee with martial prowess enemies.
The best I can see you using it is you doing hit and run tactics using the extra action of haste to Disengage and use your wings to fly away.
Is Sorcerer Gish Bad?
No I don't think it is bad but you have bad specs for it. Dex or Strength needs some attention in this matter, preferably Dex to have better AC.
Also melee damage needs that consistent source of damage from the ability score and another bonuses to make them more effective so using cantrips like Green Flame Blade or Booming Blade which stack on 2d8s on a attack you make on someone would be very helpful. Plus with sorcerrory points you can make those attacks as a bonus action via Quickened Spell metamagic to cast a one action spells as a bonus action which can allow you to either cast then strike or strike twice.
Also Green Flame Blade would meld well with the choice of Fire Dragon Ancestry cause you would be able to use your Elemental Affinity class ability add your Charisma modifier to the damage of the spell which makes you basically have the dueling fighting style with this combo.
But going for hit and run tactics it might be better to have Shocking grasp so you can escape without the need of a disengage action plus its a 3d8 cantrip at your level.