Now to the best of my knowledge Grenades are pretty deadly in 5th edition. One of the main factors in this is that there is no way to effectively avoid their very high damage.

I am here to suggest a possible houserule, an interrupt action specifically:

Fire in the hole! (-5 initiative score)
A character may choose to take evasive action in an effort to avoid a grenade or missile blast. This can mean diving behind cover, fleeing as quickly as possible or even dropping to the ground and praying. If the character is within cover they may make a reaction + intuition check with a bonus based on the cover they are behind (commonly +2 for soft cover and +4 for hard cover). Successes from this check reduce the grenade's damage before soak is rolled. A target out in the open may choose to either flee or hit the dirt. If they choose to flee they may make a running or gymnastics check. They manage to run 1 meter from the point of detonation per net hit, though at the GM's discretion obstacles or walls may limit success. The character flees directly away from the explosion, which may potentially place them in other danger. If they choose to hit the dirt they become prone but damage from the explosive is reduced by an additional 1 point per meter from the point of impact on top of the normal damage drop-off, or gives a damage dropoff of -1/m to weapons that do not normally have such a dropoff (EG flash bangs).
The 'Fire in the hole' action has no affect on the effect of gas or smoke grenades, and only a single such response can be take per action phase.

Okay, I am violating the first rule of rule design here (Specifically, K.I.S.S.) but it was either that or create three separate rules.
Can anyone suggest how I can modify this to avoid players dropping grenades at their own feet to use the flee action to increase their speed?

The main problem is the bang is really over before you have time to move unless it is on a timer.

While they may move into other dangers, giving them an instant near-teleport a few meters away will prove problematic at best and will be abused at worst.

Also remember that spells fill the area and do not drop off, so you will have to add the specific that spell aoe will not normally be reduced by this so it overrides your added caveat already about a -1/m for attacks with no drop off or someone will scream RAW and claim they can save vs fireball.

It is a good idea, but not sure if it's the best way to do it.

That's why I've thrown it out here, so those with more experience than me can pick it apart and see if anything in it is worth keeping.

My opinion - for what it is worth - is that if you are going to house rule it, and want to keep it simple, is to simply allow a defense test with a negative modifier. The hits on the defense test are simply added to the hits on the resistance test - and do not allow you to "dodge" the grenade, so it still will land on target, and you will still be resisting whatever the base DV of the grenade minus the dropoff. (Or, in the case of spells, the base DV + the number of net hits above the threshold.)

1) It is simple
2) you don't have to make up rules for movement
3) it still allows grenades to be nasty - in that if you put them on target your base damage is still really high
4) it works for both spells and grenades.

That has possibilities, but what is the negative modifier going to be? A flat number or in the case of spells would the penalty be that of the Force or something else?

Maybe the oft-repeated -2 penalty?

maybe, is the -2 really a meaningful penalty? This is an area effect after all and should be more difficult than a normal attempt.

I'd personally roll with -5 for grenades (The same as a shotgun on wide choke) and -force for spells.

That has possibilities so mooks and minions will probably take the brunt of a grenade but players will at least have a chance.