There is a new apartment complex in the Shennadoah area. I think the name is Ivy Park Luxury Apartments, but if you are working with an apartment finder website maybe you can find it. Its on Jones Creek Road and Tiger Bend Road next door to Ralph's Supermarket.
Its not close to LSU, but isn't extremely far, its takes me about 15 - 20 minutes to get to Dalrymple (sp?) / LSU Systems Office via Tiger Bend / Jeff Hwy to Seigen to I-10 to Dalryrmple if I wait until 9:30 AM when rush hour is over or leave at 6:30 before it gets bad. So if you don't mind being that distance it might be an option for you. The apartment complex is only about 1 year old and the surrounding neighborhoods are family surburban type, so its pretty quiet.
Also there is a new apartment on Seigen Lane called Bristol Place Apartment Homes that would cut about 5 - 10 miutes off the commute...
www.bristolplaceapthomes.com. Its cool because its closed to shopping, banks, the mall of La, etc. Its less than a year old.
With both of these, if the interstate is backed up, you can take Seigen all the way to LSU as an alternative route which is good (it changes names along the way to Burbank) but red lights make it longer than taking the after / before rush hour interstate. Bristolplace is on the bus route in case your car break down (or if you don't have one, you will need one in Baton Rouge by the way.. but that is a differen topic LOL) But I don't know about Ivy Park.
I hope this helps!