I am from the New England region in the U.S.
From my perspective, yes it is an accent thing but probably all English speakers (again from my own perspective) switch up the sound on the article ''a'' without even realizing it sometimes, especially in casual (and fast) conversation.
In what circumstances and how often the switch happens I'm sure depends on the regional accent, so I can't say anything authoritatively. But I'll try to reflect a little on my own usage at least.
In general, when I'm trying to be more precise and deliberate (for clarification purposes or otherwise) in my speech (specifying an object for instance) I will definitely use the /eɪ/ sound.
When speaking in the middle of a casual, average, conversation I will use the /ə/ often, and this is probably most of the time.
.....Hey, can you turn on a (/ə/) lamp for me?.....
I'm going to the store to buy a (/eɪ/) lamp and a (/eɪ/) light bulb.
I hope this helps.
Edit: This point I made about being deliberate when using the /eɪ/ sound, I think it's important to just be conscious of the specific mediums and environments that you find yourself listening to any speaker as well.
Someone speaking directly to a wide audience in a youtube video for example, may speak much more deliberately (therefore slightly differently) than they otherwise would in a casual conversation with their friend or spouse.
How someone like this speaks in their recording or stream, may not be exactly representative of how the average person on the street, where they live, speak with one another casually. Though it could be depending on the person and their style.
That being said, it will always at least be very very close to representative but as you've found these small nuances can confuse you.