Apple Mail Keeps Asking Every Week For Yahoo Mail Account(s) Passwords

Hey there. So, I have several email accounts through Yahoo mail. About three through Yahoo. Even before this latest Yahoo email hack, I've been pretty cautious, frequently changing my passwords and following ubiquitous online advice on keeping my Yahoo email accounts secure. Within the last three or four months, for some reason, my mail app on my MacBook Air and Mac Pro keeps asking me to re-enter my password within the account settings. This pop up mainly occurs when I'm not even sending or composing an email from these accounts; I would be working on something else. It's also happened when trying to send an email from that account... Invariably, a window comes up from the Apple mail app each week, asking me to re-enter my Yahoo mail password, and it takes me to the Yahoo mail account in the Apple mail app to re-enter my password. This happens for only my Yahoo email accounts, not Gmail (until recently), or iCloud, etc. Mainly Yahoo mail, and only on my Apple mail app for my Mac Pro and MacBook Air. I can log into my Yahoo email online with no issues.... Originally, I did have the two-step authentication enabled (because of the Yahoo mail hack), but since I've been having these password prompt issues, I disabled the two-step authentication on all of my Yahoo accounts. I even have the setting that allows less secure apps to access Yahoo mail enabled. Regardless, I can't seem to go three days without the Apple mail app (on my MacBook Air or Mac Pro) prompting me to re-enter my Yahoo email account passwords for my various Yahoo mail accounts in Apple mail... I have a new password for each of my Yahoo email accounts that is currently saved within the mail app settings on all of my devices (iPad, iPhone, Mac Pro and MacBook Air). The iPhone and iPad don't prompt me for my Yahoo passwords, just my MacBook Air and Mac Pro. I've double and triple checked that I've entered the passwords correctly, even trying several times removing the Yahoo accounts entirely and re-adding them back in, which Apple mail accepts my passwords again with no issue. It'll work fine again for another day or two, then I always get another prompt for the password(s). Today, I got a new prompt to re-enter my Gmail account password. Really can't figure this out. Every time I call Apple support about it, they tell me it's a Yahoo mail issue, and to disable two-step authentication and all that, which I already had.

As I've said, my Yahoo email online two-step authentication is turned off on all of my Yahoo email accounts, yet every week, I get nagged by the mail app on my MacBook Air and Mac Pro to re-enter my Yahoo email password for this or that yahoo email account. Now, it's happening with Gmail. As far as what operating system that I have installed, I have El Capitan installed on my Mac Pro, and the newest OS on my MacBook Air (Sierra, I think).

Again, my iPhone 6 Plus and iPad don't prompt me for any email passwords; that's why I'm doubting that this is a Yahoo email issue- and more of an Apple iOS mail app problem.

Does anyone know what the solution is to this problem? Sorry for the over-explanation, I'm kind of tired, and I usually get lots of people asking questions that I thought I'd answered in my original post. Anyone know how to fix this? Thanks!

Equipment I have:

11 INCH MACBOOK AIR:

Model Number: A1465

MJVM2LL/A

MJVP2LL/A

11-inch MacBook Air Part Number: Z0RL00087

2.2GHz Intel Dual-Core Core i7

Intel HD Graphics 6000

8GB 1600MHz LPDDR3 SDRAM

512GB PCIe-based Flash Storage

TBolt to Gigabit E-Net Adapter

OS: Sierra (current updated version)

APPLE COMPUTER:

MAC PRO

Mac Pro Part Number: Z0P80004V

Mac Pro Model Number: A1481

My Mac Pro Configuration:

Processor: 3.0GHz 8-Core Intel Xeon E5

Memory: 32GB 1866MHz DDR3 ECC - 4X8GB

Startup Disk: 1TB PCIe-based Flash Storage

Graphics: Dual AMD FirePro D500-3GB VRAM

OS: El Capitan (version 10.11.6)

IPAD AIR 2:

Pad Air 2 Part Number: A1567

MH312LL/A

IPHONE 6 Plus

DEVICE MODEL: APPLE IPHONE 6 PLUS - 5.5"

Phone 6 Plus Model Number (128 gig): MGAP2LL/A