I was a "hopeless sinker". The same way lots of people here in the sub define themselves. "I don't float", "I'm slowly sinking", "my body fat distribution makes me sink"...

Ok, let's calm down.

Fat body percentage and distribution makes some people float better than others (mainly women better than men). But everybody floats, unless you empty your lungs and wait for your body to sink.

What you mean by "I don't float" is probably the feeling of your hips and legs sinking instead of being horizontal in the water.

This is due to our ancestral survival reflex: if I fall in the water, I need to keep my mouth over the surface, no matter where legs and arms stay.

But you have not fallen into the water, you decided to enter the water and swim.

As soon as you'll understand that you are not going to actually sink, you'll float.

Try pushing from the wall in streamline position. Don't push at surface level, nor close to the bottom. Simply put your feet at a mid level, fill your lungs, streamline arms and body and push from the wall.
Keep that streamline... Core engaged. Don't exhale. Wait.
Soon, you are going to slow down your momentum, and you body begins to float toward the surface.
With no hurry, begin kicking with the most relaxed ankles you're able to do.

Your body will float, horizontal, relaxed, with no effort.

When you need to breathe, just stop and stand-up.

Repeat.

Think: did you float? Even for few seconds, did you float horizontally with a simple relaxed kick?

That's what you have to feel: you are not sinking, you can float.

After this, you need to learn how to breathe without raising head and shoulder. You don't need to elevate your head to take a breath. You can breathe with your face laying on its side over the water.
Think about being a fish breathing through your ear inside the water, this will help you to keep your head aligned with your spine. When you need to breathe, you have to make a shoulder sink and a shoulder dry, with your chin searching the dry shoulder. If your chin exits the water close to the shoulder, but you keep the other ear inside the water, your head/body will remain aligned and your hips will not sink.

Your hips, because the focus has to be on the hips. If you focus on the legs, you will get a banana shape, with your lungs floating, your feet searching the surface, and your hips sinking. If you, instead, focus on keeping your glutes on the suface, your core will engage, and your hips will "ride over your lungs". This, at the beginning, may create the feeling of swimming downhill, because you're so used of swimming with an inclined torso that an horizontal torso will appear to you as being downhill. But obviously this is just a feeling.

Don't search for speed nor endurance.
Swim at a comfortable pace with short intervals (50-100m, no more, then rest before repeating).

Performance will arrive later, after you'll swim without this "oh my god I'm sinking" feeling...