It is very viable
On most levels, such a multiclass is stronger than a single-class paladin.
There are very strong synergies between the classes, patching each other's shortcomings.
Paladin problems solved by Sorcerer levels
Lacking good ranged options
Paladins are quite ineffective against flying enemies. They can't spare a decent Dex for bows, and can only throw one javelin per Attack1.
Levels as a Sorcerer can give you cantrips like Fire Bolt that can outperform both.
Lacking good options if you have disadvantage
If you are poisoned or restrained (but not blinded2) it is usually better for your DPR to use a cantrip that does not require an attack roll.
Unused bonus actions
Unless the Paladin takes Shield Master or Polearm Master, he has no easy and efficient way to spend bonus actions3.
Sorcerer 3 can turn any spell into a bonus action with Quicken Spell.
Few spell slots
Between Divine Smite and usual spellcasting, Paladins could always use more slots.
Sorcerer levels provide more, but 1 level is not enough.
Paladin X/Sorcerer 1 has the exact same slots as a Paladin X+1.
P12/S8 has 14 caster levels, 4 more than P20.
Limited spell selection
Only Rangers have a smaller selection than Paladins.
Sorcerer 1 gives you 4 cantrips, and some great spells like Shield, Disguise Self, or Absorb Elements.
Few spells prepared
Even on level 20 and with Cha 20, a Paladin can only prepare 15 spells freely4.
While spells known and spells prepared are not the same thing, Sorcerer still adds to your options.
P12/S8 starts the day with 20 spells, 5 more than P20.
Sorcerer problems solved by Paladin levels
Sorcerers have to spend a spell slot to achieve an acceptable AC and have the worst HP among all classes (on par with wizards). Dragon Sorcerers do a bit better.
Paladins can have a better AC without spending a slot, and add a lot to your hit points.
Small damage output without spending resources
Cantrips are much less damaging than weapon attacks, even for Dragon Sorcerers.
This is especially painful with Quicken Spell.
Paladins do great damage round after round even without spending any resources. After a Quickened spell you can still use a very efficient Attack action.
Paladins can cast most of their spells while holding a weapon and a shield. This does not work with Sorcerer spells.
Your best option is using a two-handed weapon without a shield.
No high-level spells
Paladins get their 5th level spells on level 17, but a 10th level Bard can take any of them. If they are not overpowered on level 10, they are definitely weak on 17.
Not much of a loss.
P6 is a great point for multiclassing, you already have the best three Paladin features (Divine Smite, Extra Attack and Aura of Protection) and the next best (Improved Divine Smite) is very far away.
As you said Shield is great, and the 4 cantrips are also very useful, especially Green Flame Blade5.
I would take 5 level of Sorcerer right away, Quickened Haste is that awesome.
Then 6 more levels in Paladin, for Improved Divine Smite and the ASIs. Or, if you have GWM, just 2 more levels of Paladin for the ASI.
The rest in Sorcerer.
In my opinion, this build is stronger on every level than a simple Paladin. I would only miss Unyielding Spirit, but a Quickened GFB makes up for it.
Usually, I recommend taking the first Sorcerer level earlier, after Paladin 1 or 2. Starting with Sorcerer would take away Heavy armor and 4 hit points.
This build is behind a single-class Paladin on level 5, possibly even 6, but I really like the flexibility and the ranged options.
1) Actually, they can throw more than one, but they can not grab it, having spent their free object interaction on grabbing the first one.
2) For most cantrips that require a saving throw you need to see the target.
3) I am not a fan of Smite spells, their damage is below that of a Divine Smite with the same slot and they use up your concentration. I would much rather concentrate on Bless, Magic Weapon, or Haste.
4) On 20th level you also get 10 predefined Oath Spells, the usefulness of those is very uneven.
5) A Quickened GFB on level 11 does more damage than your 2 usual attacks combined.