I know a lad that got into Oxford last year who wrote his personal statement like you have presented yours, without lines between the paragraphs. It may look nicer with lines between the paragraphs, but it isn't necessary - even if admissions tutors prefer it. If an admission tutor makes the decision on whether to offer you a place or not purely down to how you have spaced your paragraphs, then perhaps you shouldn't be going to that uni? It is their job, is it not, to read the contents of somebody's personal statement to make the decision, not whether they have spacing in their paragraphs to make that decision.
If I were an admissions tutor, I would ask why there are lines between paragraphs. It would suggest that there is an inability to talk at great length about your subject and why you want to study it. It would actually put doubt in my mind about the applicant's suitability for the course. You can argue both ways against "lines or no lines".
Indeed, I have just looked at a copy of my personal statement from last year that had no line breaks between paragraphs. It looks fine. There is clearly paragraphs in the text, and its not too hard on the eye. I really don't see the problem. If you don't use paragraphs at all, however, then perhaps you should go back and do your English GCSE again. lol