What quotes from Romeo and Juliet show that Romeo and Juliet were trying to escape their present troubles by stating they are in love with each other? For example, Romeo is trying to get over a girl and Juliet is trying to avoid marrying someone her parents want her to marry. What quotes show this?

This is an interesting interpretation. I think it's a mistake to come up with an idea first and then try to retroactively fit the facts to the theory, but it seems you have already spotted elements in the text which you think point towards this reading.
There aren't any quotations which decisively "show" this idea to be true, as such, but I can think of some aspects which might bolster your argument. For example, Benvolio certainly thinks the best way for Romeo to get over his attachment to Rosaline is to latch onto another girl:

Go thither; and, with unattainted eye,Compare her [Rosamund's] face with some that I shall show,And I will make thee think thy swan a crow.

Evidently, Benvolio thinks Romeo is attached to Rosaline without much hope of ever attaining her—and also, he thinks Romeo could do better. Romeo is very certain that he will not get over her: "the all-seeing sun / Ne'er saw her match." He is also, indeed, unhappy with his attachment to Rosaline, believing it to be love and stating that "it pricks like thorn."
I'm not sure that we can argue that Romeo is trying to escape his love for Rosaline by falling in love with Juliet, as such. Certainly that is what his friends hope for him, but as a man who is so disillusioned with love, it's hard to believe he'd want to escape it by rushing headlong into yet another love affair. Rather, perhaps we can argue that Romeo doesn't really love Juliet because he is so fickle—he speaks as if his love for Rosaline is undying one minute, and the next is declaring his love for Juliet:

Did my heart love till now? forswear it, sight!For I ne'er saw true beauty till this night.

One moment, Rosaline is the fairest thing he has ever seen; the next, he is head over heels for a girl he has never even spoken to and declaring that his love for Rosaline was not real love. But is it, then, really love that he feels for Juliet, or is it a case of a young man moving on too speedily and recklessly to the next girl? Even Juliet cautions Romeo against his haste, when he declares that "love's light wings" have helped him into her orchard—"If they do see thee, they will murder thee."
As for Juliet seeking to escape her marriage with Paris, I'm not sure there's evidence that she necessarily dislikes Paris, as such—she does not seem particularly eager to marry at fourteen, but on the other hand, is happy to marry Romeo. When Capulet says that he will "drag [her] on a hurdle" to marry Paris, Juliet resists ("I beseech you on my knees"), not out of personal loathing of Paris, but because she has already declared herself to Romeo. Possibly she simply wants to make her own decisions.
We can also interpret that the young lovers may rush into marriage for another reason—in this time period, they would not have been able to sleep together outside of marriage. So, is it lust that drives Juliet into hasty marriage? Juliet is eager for night to

learn me how to lose a winning match,Play'd for a pair of stainless maidenhoods:Hood my unmann'd blood, bating in my cheeks,With thy black mantle.

I'm sure you can pick out some other elements in the text yourself, but this should offer you a place to start.