I have to write a letter of intent to a college stating why I am a good fit for their journalism program. In the rough draft, I wrote this sentence:

"I follow current events on a casual, weekly basis."
How can I sound a little more dedicated to my approach in this statement?

Leave out "casual."

I left out casual in the sentence, and it looks like a sentence fragment.

"I follow current events on a weekly basis. The kind of news that I would pay attention to would be the weather, natural disasters, sports, and most tragic events.
A couple of tragedies have made an impact on my life, such as the Sandy Hook Elementary School shooting that occurred last December, and the September 11th attacks in 2001. I remember the aftermath of 9/11 like it was yesterday because it was devastating to watch and hear about people die from the World Trade Center in New York City, The Pentagon in Washington, D.C, and in rural Pennsylvania. The Sandy Hook tragedy made an impact ... 20 little children were shot to death multiple times by a gunman who was speculated to have Asperger’s Syndrome, and I have that disability. I can recall the story of Victoria Soto, when she was praised as a hero for saving her students’ lives in the face of danger. I keep track of professional and some amateur sports (ex. hockey, baseball, football, soccer, etc.) by watching most sports highlights on TV and the Internet. The reason why I pay attention to sports is because ..."
The paragraph needs more work

What does a sentence fragment look like??

That first sentence is fine. It has a subject and a verb and is a complete thought. "looks like" is not a reliable way to judge whether a sentence is complete or incomplete!!!

To sound more dedicated to your approach in the statement, you can try the following revisions:

1. Increase frequency: Instead of stating that you follow current events on a "casual, weekly basis," you can mention that you stay updated on current events on a "daily" or "near-daily" basis. This demonstrates a higher level of dedication to staying informed.

2. Demonstrate active engagement: Highlight your active involvement in following current events. You can mention that you actively seek out news sources, analyze different perspectives, or engage in conversations or debates related to current events. This shows your dedication to gaining a deeper understanding of the subjects.

3. Show varied sources: Instead of just stating that you follow current events, you can specify that you follow them through a wide range of sources such as newspapers, online platforms, podcasts, or documentaries. This showcases both your commitment to staying informed and your ability to gather information from different mediums.

Revised sentence: "I am deeply committed to staying updated on current events, immersing myself in their analysis and implications on a near-daily basis. I actively engage with various news sources, including newspapers, online platforms, podcasts, and documentaries, in order to gain a comprehensive understanding of the ever-evolving world around us."