You can make your save before losing your move...

From the text of the spell (PHB 264, emphasis mine):

A dancing creature must use all its movement to dance without leaving its space...

Note the term "use" rather than something like "lose" or "have its speed reduced to 0". The spell compels you to move your speed within a 5 foot square. Fortunately, from the description of a player's turn (PHB 189):

You decide whether to move first or take your action first.

Even better, if slightly contradictory (PHB 190):

You can break up your movement on your turn, using some of your speed before and after your action.

In this edition, players are given a great deal of latitude on when to use their movement. Since dancing is a use of that movement, you can choose to save before using it up.

It helps to think about what limits your movement speed under regular circumstances. Remember that a round, and thus a turn, lasts ~ 6 seconds. When your move speed is 30 feet, it doesn't mean you are out of breath after running 30 feet, it means that it takes you about 6 seconds to move that distance with enough time left over to do something else. Dancing keeps you from moving anywhere as the clock ticks down, but if you make your save quickly, nothing in the wording of the spell suggests you would be prevented from using the remaining time in your turn to move your speed.

...but you still dance

You start dancing as soon as the spell is cast, with all of the attendant detriments (disadvantage on attacks and Dexterity saves, granting advantage to attackers). There isn't much you can do about that until the start of your turn, but at least you don't have to stay immobilized.

RAW - No, Otto's Irresistible Dance is not a Charm spell

It does not say that the target will be "charmed", and the condition it imposes is significantly different than the description of the "Charmed" condition.

OID:

A dancing creature must use all its movement to dance without leaving its space and has disadvantage on Dexterity saving throws and attack rolls. While the target is affected by this spell, other creatures have advantage on attack rolls against it.

Charmed:

A charmed creature can't attack the charmer or target the charmer with harmful abilities or magical effects. The charmer has advantage on any ability check to interact socially with the creature.

A dancing creature can attack the caster (with disadvantage -- and that disadvantage applies to any attack, not just against the caster) and can cast AoE spells that include the caster and anyone else they choose in the target area with no penalty. They just can't move freely. A charmed character can move freely but can't attack or target the caster.

Furthermore, if it were a Charm spell, it would not be necessary to specify that "[c]reatures that can't be charmed are immune to this spell." That appears to be listed as a special exception.

RAI - No, Otto's Irresistible Dance does not impose the Charmed condition

Jeremy Crawford has tweeted saying:

Being charmed means being subjected to the charmed condition

Since OID does not impose the Charmed condition, the Fey Ancestry resistance to "being charmed" does not apply.

What makes sense?

Webster's dictionary says:

charm: to affect by or as if by magic : compel

Magically forcing someone to dance seems to fall completely within that definition.

I would give those with Fey Ancestry advantage on saves against all enchantment spells that compel the user to do something that is not their own free choice. This includes many spells that do not specifically say they are "charms" or that they impose the "Charmed" condition, such as command, compelled duel, and yes, Otto's irresistible dance. I believe those all fall within the common sense and dictionary definitions of "charmed" in this context, and there's nothing in the description of Fey Ancestry that says it only applies to things that grant the "Charmed" condition.