I dislike the idea of my eladrin having four appearances and have been seeking a backstory that would solve this problem.  One idea is for the character to have no emotions.  However, this raises the question of how mechanically the character could change his fey step ability since this is linked to his seasonal affinity, which is tied to his emotional state.  

One possibility is that the character suffers from alexithymia, which as I understand it is a condition in which an individual has difficulty recognizing and expressing emotions (and reacting appropriately to the emotions of others).  Perhaps the character is aware when he is feeling an emotion, but he cannot discern what it is and does not express it.  Being unable to express the emotion, his appearance does not change.  However, the actual fey step ability is linked to the emotion that is felt, whether it is understood or not.  

Any other ideas to to address the issue would be appreciated.

This doesn't really seem like a rules question, so it should probably be in Tips & Tactics or maybe even Story & Lore?

What I will say is that while the updated Eladrin weirdly emphasises the mood more (whereas other updates tend towards being more permissive), in the legacy version it was always a lot more optional; i.e- the seasonal affinity can affect or reflect mood, but it doesn't have to. There are still some lore hooks if you want to be really strict though; an eladrin doesn't need to shift seasons at all, some can and will stay a single season their entire lives, or only shift as children, while others might shift for other reasons.

For example, they might shift with the actual seasons, so they're autumn in autumn, spring in spring etc., you could also take cues from the environment, for example in a hot environment you shift to summer, in cold winter etc. Since the season is tied to trancing you could even simply have a vision of a particular season, and that's what you are when the trance ends?

Wouldn't an Eladrin without emotions just be a Shadar-Kai?

Also, some Eladrin never change their season their whole life, but that doesn't mean they don't feel other emotions, they just don't wear their mood visible for everyone.

Instead of contriving an emotionless character, if you don't want to change between forms I would just pick a form and say "yeah this is just an autumn Eladrin". Unless the lore for eladrin is very important to your campaign world, feel free to play a little loose with it.

The Astral Elf (Spelljammer) and the Shadar Kai (Mordenkainen's Monsters of the Multiverse) both have nearly the same mechanics as the Eladrin, but don't change seasons.  So either of those sounds like it would be a better fit for your concept than the Eladrin as written.  Of course, you could simply be an Eladrin who doesn't change seasons, having a personality trait that's so dominant, that other emotions you feel aren't sufficient to change your season.

I appreciate all the responses thus far. 

The problem with staying in one season is that it diminishes much of the value of playing an eladrin vis-a-vis an astral elf or shadar-kai.  The utility of the eladrin is tied to its access to four separate fey step abilities.  If I stay in one seasonal affinity all the time, then I lose this value.  While not all of the fey step abilities are equally valuable (or valuable at all), the autumn, spring, and arguably winter affinity abilities are.  Autumn is a standout, though the associated mood (all sweetness and baking cookies like grandma) is antithetical to the character I want to play.  I am also not interested in playing a goth, and frankly dislike the Raven Queen, so shadar-kai is not a preferred alternative.  There are also other problems with shadar-kai, including a lack of clarity on appropriate languages other than common and elvish and a singularity in appearance (grey skin, black eyes).

Another, perhaps simpler idea would be to make the character an albino.  If the character has albinism, then he should not be able to change his appearance.  Regardless of the seasonal affinity, he would be white skin, shite hair, and red (or blue/violet) eyes.   

As I was thinking about this, I Googled "emotionless eladrin" and found a homebrewed dusk eladrin:  https://imgur.com/gallery/hqao0ZV.  

Quote from Xeyd >>

I dislike the idea of my eladrin having four appearances and have been seeking a backstory that would solve this problem.  One idea is for the character to have no emotions.  However, this raises the question of how mechanically the character could change his fey step ability since this is linked to his seasonal affinity, which is tied to his emotional state.  

 Pick a different race?

Why be an Eladrin if you want to homebrew away its signature feature?