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I've watched you agonize over word choices and I have no doubt you'll get your credentials, based on perseverance alone. I'm so glad you included the whole paragraph. It helps someone like me understand the flow of each sentence when grouped into a mass; it makes a difference. Based on this, I've made changes intended to improve flow.

I noticed that the word "solve" or "solutions" was included three times in the first three sentences. Shortening the third sentence works, I think; that way, you have a few long sentences, then a short one, then long (this is good from the reader's perspective, and helps it flow). In the first sentence I changed "the appropriate step" to "an appropriate step" because there are other things that can be done to resolve societal problems (but I won't feel bad if you change the wording back). Anyway, your message is clear, so I hope you won't have to agonize much longer!

"I view a career in law as an appropriate step to help resolve conflict in society. I have the ability to find effective solutions for problems by applying consistent listening and communication skills. In addition, I enjoy puzzling together the facts of a situation and finding any missing links. Too, I have the strong sense of perseverance that is essential to achieving short-term and long-term goals. Through hard work and dedication, I look forward to a career within the area of law."

The second one. Make problem plural. "to find effective solutions for problems"