The Abir Yaakov is the grandfather of the Baba Sali and great grandfather of Rav David Abuchatzeira. Here's what he writes in regards to our struggle: 

"The rectification of the brit (lies in) pursuing peace (in) Yisrael...because the defilement...causes precious sparks of holiness to be scattered...To the degree that these parks are scattered, so too are the holy Jewish people scattered...As we work to gather the Jewish people back together in peace, love, and unity, so too does Hashem gather together the holy sparks that we scattered...and returns them to their holy source in Heaven. For this reason it is so important not to feed the disunity and strife that rages among us. Especially damaging in this regard are the sins of lashon hara. The brit of speech parallels the brit milah. We guard the brit of speech by refraining from lashon hara, thereby correcting the sins of brit milah...As a person struggles to improve himself in one area, he improves in the other as well." (Teachings of the Abir Yaakov Vol 2 Chapter 1 Page 19) 

I myself am studying the laws of shmirat halashon and I recommend it to all of you as well. The truth is so much we say thinking it's permissible is not actually permissible and actually lashon hara or rechilus. This came to my attention only after studying the hilchot lashon hara and it's something that concerned me greatly. Rabbi Avi Wissenfield on TorahAnytime, Rosh Yeshiva of Beit Dovid, has classes on hilchot lashon hara from Sefer Chofetz Chaim that are usually less than 10 minutes and I recommend watching them. Obviously this is a daily commitment sort of schedule but it's pretty manageable for me.  

(If somebody wrote about this before me, I didn't see and I'm sorry)

So you're saying since there's a connection, studying lashon hara can help? 

Yea, that’s what I took away from it. 

Very interesting. I actually find it hard to work on both at the same time and find I need to prioritize. But maybe not.

I don't really think so. אדרנה, we have a self-control "muscle" that keeps the יצר הרע in check. Exercising it in שמירת הלשון will strengthen it for שמירת העינים and שמירת הברית. 

Add to that the exciting שמחה  that accompanies the satisfaction of withstanding a נסיון and you are come out ahead בכל הענינים  

I think. 

The Talmidei Baal Shem Tov bring this concept many times in their sforim - that the bris ha'lashon is parallel with the bris kodesh, and that zehirus in speech assists one in shemiras ha'bris. It is a metzius in the way we were created.  So although working on two things at once is often not advisable, this seems to be an exception.