If you want to find me, then here's my suggestion:
You must look in a barnyard and forget the question.
We start with the word "barnyard". The "question" in the word barnyard is the letter Y (Why), so we start by removing that and we have BARN ARD. Credit to @Jafe for solving this.
Fill the gap with yourself, but do not change the sound
Yourself is the letter "I". So we insert this letter into the gap left by the Y! "Do not change the sound" refers to the letter "Y" which can also have the same sound as "I", so in fact the "I" replaces it leaving BARNIARD. Also, in the final answer "drain" the A+I make a Y sound.
And then move your eyes west and I will be found.
Move your eyes west either means read the entire word from RIGHT to LEFT (thanks @jafe) now the word reads: DRAINRAB. (Even before swapping the Y for I you would have got DRAYN which sounds right - confirming that the AI make the sound of the Y)
But wait — my memory's not straight, rather curved!
Remember! The full word was BARNYARD, so we need to look back at the complete word (or BARNIARD after we swapped the Y for I). This may also refer to the shape of a drainpipe (which is curved, like an 'S-bend', or the 'trap' on a basin or bathtub.
Please order the take-out where refreshments are served.
Refreshments are served in a BAR so we "take-out" the letters B-A-R from the entire word and we are left just the letters DRAIN.
It should all be complete, but my head is all soaked.
RAIN would soak your head, telling us to rearrange the letters this way, but that isn't the complete word. Also the hint points to a drain that would be found in a shower, which soaks your head.
Sometimes there's dirt; sometimes there's smoke!
This is the clue to the word "Drain". Drain pipes are dirty. And apparently smoke can be used to test drain pipes (borrowed from @SMR, who borrowed the answer from me)
Now the crosses have turned — it is time for the running;
The "crosses have turned" refers to taps (faucets for Americans), which often have cross-heads, being turned on. Now water is "running" and going down the DRAIN.
Bubbles are bubbling, and droplets are drumming.
Now that the taps are turned on you are in the shower, enjoying the bubbles and hearing the sound of the water droplets drumming, as water disappears down the DRAIN. (also your hint mentions the shower)
The title: Another New Riddle I Designed
The lead letters A-N-R-I-D are also an anagram of DRAIN. (pointed out by @SMR)