This is basically M.O.R.E from Undeadsewer (original mod: https://modworkshop.net/mod/1497), but with fixes applied (from https://modworkshop.net/mod/21544, credit to the creators of that version because i mainly only fixed and polished those updates),
I went ahead and kept the Extras module out as melee breaching doesn't want to work anymore, and it's kinda cheaty anyway,
but I managed to extract the Melee Sprinting module from the original mod and added it as an optional download. Just put that in as well if you want to be able to sprint while charging your melee weapon.

The original updated version had some yank when it came to running and shooting after a melee attack, as well as running while reloading, which cancelled your reload animation, but kept the reload going. It was also easily possible to get your melee weapon stuck in your hands and being able to shoot from it (which was kinda hilarious ngl). Also, if you had the ability to run and shoot, the normal run animation would play after meleeing while sprinting, which also looked very off.
I think I managed to fix these issues, at least on my end it works like a charm now, but please let me know if it works for you guys too. I didn't test that toroughly except for my modded setup.

Only visual issue still pertaining is that with some slow hitting melee weapons, your gun comes out for like a split second after 2 melee hits, but I really don't know how to fix that. Please let me know if you got any ideas about that! Better Melee Animation by Hinaomi (https://modworkshop.net/mod/26768) fixes it for some of them tho.

It's also compatible with Advanced Movement by Solo Queue Pixy (https://modworkshop.net/mod/24736), and really adds to the features of that mod, but should also work fine without it.

For those that aren't aware, Melee Overhaul REvamped allows you to hold your melee weapon in your hand, for better and more reliable melee combat. It also adds melee dismemberment, hit effects on surfaces, some additional effects on weapons like the electric brass knuckles and the option to turn off player head movement and zoom while charging/holding melee. All features are toggleable/tweakable in the options menu of the mod, except for melee sprinting which is under it's own menu.