Hey! Great stuff! First rides are always a learning experience. The KLR is a great bike, cheap, reliable and capable of taking your guitar and a ton of stuff on board.
But since you said off road riding is not a priority, then this opens up a ton of other bikes too. Basically, any street bike can do this ride. But the KLR is well suited to carrying camping gear and stuff as it sits. And by the way, your early dirt riding experience is quite valuable actually, even if it happened when you were very young. The memories are still there.
A Suzuki DR650SE (from 1996 to present are ALL good) dual sport is also very inexpensive and very good travel bike. Simple, cheap to maintain, reliable as a hammer. Needs a better seat. Also look around at older dual sport and standard bikes too. Lots to choose from.
The first thing I would do now is to take a MSF rider training course. You don't need a bike for the course, they provide one. This is a basic riding course that is really fun and can save your butt. Teaches good habits and breaks old ones. Essential. Motorcycle riding is dangerous, be prepared. Learn how to ride and ride well.
Here is a short list of other potential bikes:
Suzuki SV650 (best!)
Suzuki GS500E (almost any year. Older OK)
Kawasaki Ninja 500R
Kawasaki Ninja 600R (older one OK)
Honda Nighthawk 750 (95 up)
Honda CBR600R (older back to mid 90's)
Check Craig's list and be ready with cash to buy. The really good deals will go quickly. The San Fran Bay area is better for buys and numbers of bikes is better than LA area. Lots of good bikes up north and much closer together.