I heard some advice a while back that was along the lines of, "stop apologizing and start thanking," and I feel like it's positively impacted how I phrase things.

Instead of asking forgiveness and moving the conversation into them feeling they have to defend their values on the spot, showing gratitude for their understanding actually makes people feel more valued. "Thank you for your patience" is an entirely different vibe than "sorry I didn't get back to you" and puts much less burden on them. It shows you care about their time without making the focus about your failings and whether or not they agree they are failings.

It's subtle, but I find it's made a huge difference for me.

I also agree with others, in my experience apologies should be reserved for regret and actual feelings of penitence. It's actually a very strong value of mine nowadays, and it certainly makes me much healthier.

Just some thoughts about what I've learned about this particular situation, it's up to you how valid you think they are.

The frequent over apologizing was a thing I had issues with as well. I focused on stopping because I realized it can be negative for those you communicate with.

Apologies are hard to reply too. So if your apologizing all the time your are putting a burden on them to acknowledge it. This can strain communication over time.

Also, like other things, overuse of apologies weakens them. If your a little more picky about when to apologize it will make them more genuine. This is also why I avoid yelling and cursing. So when I do those things, the expression has some potency.

Lasty it's unprofessional in written communications. This is probably a more controversial opinion. But for the reasons above and because it can make your communication less concise. In the professional environment being clear and concise is more important then being polite. (Though you still need to be polite).

The added bonus is that doing this will make you sound more confident.

I am also quiet charming. Mostly by being unspoken.

To me it's not actually an apology

Then stop doing it. An apology is for when you did or could have fucked something up. "I'm sorry I haven't been able to reply sooner" means you are aware of a deadline involved for the other party and that deadline may have been fucked by your untimely response.

when they reassure me I don't need to apologize

This means there was either no deadline involved or the deadline did not pass or nothing was fucked as a result. Also means that you may have incorrectly assumed information about the existence or properties of a deadline.

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Knowledge and skill level can be divided approximately by degree and type of brainpower used. In this manner, Unconscious Use > Conscious Use.

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Think about it this way: You've blocked everything out so you can dice an onion perfectly. If one thing distracts you, the dicing is gonna get fucked. Meanwhile, Sous Chef-san can dice an onion with his eyes closed and yelling at the new guy because he's about to dump the wrong spice into the soup. Who one do you think is better at chopping onions?

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Memorization is not learning. Learning is the ability to cross-reference information, and intelligence is the ability to select the appropriate knowledge to apply for the situation.