No.

The simplest way to approach this is to take the rules literally. The interplanar effects only happen when the sphere touches a planar portal or an extradimensional space. An antimagic field is none of these, so it will not "go off".

Given that there's no specific interaction between an antimagic field and a sphere of annihilation, we then fall back on your first interpretation, which is that the sphere winks out of existence only while the antimagic field is active, in the same way that any other magic item is deactivated.

Being a wizard definitely expands your options. A quick recap of the sphere movement rules for posterity:

The Sphere is stationary until someone controls it. If you are within 60 feet of an uncontrolled Sphere, you can use an action to make a DC 25 Intelligence (Arcana) check. On a success, the Sphere levitates in one direction of your choice, up to a number of feet equal to 5 x your Intelligence modifier (minimum 5 feet). On a failure, the Sphere moves 10 feet toward you. A creature whose space the Sphere enters must succeed on a DC 13 Dexterity saving throw or be touched by it, taking 4d10 force damage.

What to take from this:

1. Any creature with 60 ft. of the sphere can attempt to move the sphere
2. Any creature without training in Arcana or a high Intelligence is very likely (or certain) to fail
3. Failing still moves the sphere (10 ft. toward creature attempting to move it)

My assertion: failing to control the sphere and having it move 10 ft. toward you does not constitute having control over the sphere (since it doesn't respond to your wishes), so it would remain 'uncontrolled' by definition. This is debatable, as it's not explicitly laid out in the text for the sphere, but I think it makes sense. If anyone can find precedent that this specifically does NOT work, let me know.

Taking the assertion as true, you can have multiple creatures attempt and fail to control the sphere in the same turn as a means of moving it. By creating a chain of creatures focusing on this task, you should be able to move the sphere up to the speed of the creatures involved without any of them being annihilated.

The sphere can only move as fast as your 'train' can move (to avoid having the sphere destroy the creatures), so assuming you had six creatures with a fly speed of 60 feet of the non-simulacrum variety, you can have your creatures arranged as such (position is the X coordinate on a grid):

Sphere: 0
Cre1: 12
Cre2: 14
Cre3: 16
Cre4: 18
Cre5: 20
Cre6: 22

Your turn:
Cre1 moves sphere to 2 as an action, then flies to 24
Cre2 moves sphere to 4 as an action, then flies to 26
Cre3 moves sphere to 6 as an action, then flies to 28
Cre4 moves sphere to 8 as an action, then flies to 30
Cre5 moves sphere to 10 as an action, then flies to 32
Cre6 moves sphere to 12 as an action, then flies to 34

In this way the sphere would move 60 ft. per turn with no action on your part (just an initial instruction to the creatures to perform in this manner until instructed otherwise). Obviously this exact example only works in wide open areas so the creatures can move freely, although you can have the creatures move around corners and such in a similar fashion.

Sphere: 0, 0
Cre1: 0, 4
Cre2: 0, 5
Cre3: 0, 6
Cre4: 4, 6
Cre5: 6, 6
Cre6: 8, 6

Your turn:
Cre1 moves sphere to 0, 2 and moves to 6, 8
Cre2 moves sphere to 0, 4 and moves to 6, 10
Cre3 moves sphere to 0, 6 and moves to 6, 12
Cre4 moves sphere to 2, 6 and moves to 6, 14
Cre5 moves sphere to 4, 6 and moves to 6, 16
Cre6 moves sphere to 6, 6 and moves to 6, 18

Lower level wizards can use Animate Dead cast in a 4th level slot to create and maintain 3 skeletons (speed 30 ft.) which would work about as well, although the sphere would move at 30 ft. per turn instead, skeletons can't fly, and it gets a little trickier with the skeletons' initiative, but it's still not bad.

Sphere: 0
Skel1: 6
Skel2: 8
Skel3: 10

Skel1's turn:
Skell1 moves sphere to 2 as an action, then walks to 12

Skel2's turn:
Skell2 moves sphere to 4 as an action, then walks to 14

Skel3's turn:
Skell3 moves sphere to 6 as an action, then walks to 16

The only caveat is the skeletons not all acting simultaneously, so you might end up with unexpected behavior if one gets killed, etc.

The only other realistic option I can think of would be to use Geas, although this moves into the category of DM fiat in whether or not the hapless peasants would comply, screw up because they're stupid peasants, or just willingly die, etc., since you don't have absolute control over them. A number of other summoning/animation spells exist, but none that have a reasonable duration that I know of (Animate Objects or Conjure Animals, for example).

So here are the best options so far as I can see:

• Best: 7th level wizard with Animate Dead in a 4th level slot
• 2nd Best: 9th level wizard with Geas and 3 hapless peasants (DM can screw with it)

Of course you could create a simulacrum of yourself, have them create one, etc. and have them True Polymorph themselves into owls! This does NOT work in AL , though, Sansuri's Simulacrum or otherwise.

Joke answer: Hire 3 peasants 1gp per day to do this