1000 words is about 3-5 main points.
Paragraph structure:
opening sentence refers back to the preceding paragraph and looks forward to what is going to be in this paragraph: "Having discussed blah/having outlined the main argument of this essay .... I will not look at blah blah ..."
Next however many sentences: dicuss blah blah
Last sentence of paragraph: Recap briefly what you did in the paragraph and look forward. "In conclusion/to sum up .... pithy summing up of point .... this raises the question of (whatever next point is)... there is also the issue of (next point) .... I now turn to (whatever you turn to in your next paragraph).
That's a rough guide. Obviously, you need to vary it a bit because you can't have all your paragraphs beginning and ending that way - it would read very strangely. But that, in principle, is how it flows: Your essay has an intro and a conclusion and in the middle is the main bulk (your points). You flow from paragraph to paragraph, point to point, with orientating sentences at the start and end of paragraphs basically telling your reader how this paragraph relates to the one before, what is going to be in that paragraph, and how it relates to what comes next.
1000 words isn't enough to be very flashy - if you have a lot of material, it might be better to just dump it all, paragraph by paragraph.
I sometimes write essays by writing a paragraph about each point, then deciding on the order, then writing the linking paragraphs, and then writing the introduction and conclusion.
Not sure this matters for a 1000 word essay (you probably don't have enough space):
Introduction = describing what I've written/what the reader is going to read (kind of like a map) + sometimes an explanation why it's in that order; conclusion = recap of what they've read and why I think it's important.
Last point: Writing the essay is a question of getting words out on paper, it's the editing that is the glossy bit.
What about just writing the whole thing quickly and then spending more time, after that, doing the glossing and buffing and shining? I think you might find it easier to add the 'flow' when you have a clear idea of what you're writing + the words to say it already (in most part) written.