. What it feels like is that when there are a lot of thoughts, there is less direct perception of reality. Of course thoughts are also part of reality but it seems like they veil the experience of sight, sound, etc
Maybe it's where the attention is? We often say 'lost in thought' and it can be like that. Whatever story is going on, whatever thoughts are 'saying' can seem to be reality, to be actually happening.
A good way to see the difference between what's really happening and what is only imagined is to go and find a piece of fruit, an oange or an apple maybe. Place it so you can't see it and, with eyes closed, for a minute, imagine that fruit. Imagine it vividly. It's colour, texture, scent and so on.
Now fetch the real fruit and hold it. See its colours, it's scent and even take a bite.
Now compare the experiences. The thought-fruit with the actual fruit. Is there a difference?
. but it still feels like there is some kind of veil in between the perception and that which perceives.
Sure. What is that veil? (and notice that 'veil' is a concept that already suggests some kind of physical or 'real' layer... Whereas it's just an idea.
This is where you need to look. Examine the actual experience of hearing, seeing, touch, taste, smell. Yes thoughts are likely to present commentary but that's just words and mental images. You're concerned with the immediate experience of sensation. That's actually happening whereas whatever is going on within thoughts is a kind of abstraction.
. . I don't really know what either of those things actually are, and I don't even know if it's a false idea that there can even be no separation.
Take a red cup, for example. Yours may be white or whatever but find an object with a clear single colour and put it in front of you. Now, with no particular effort at all, notice that redness. Now, is the redness 'separate' from the seeing of it? Is there any way of prizing apart the redness of the 'cup' from the 'experience of redness'?