The 737 isn't going to help you get a job, but it does help with a couple of things. It helps acclimate you to a civilian training process, believe me, it's different than military training. You'll adjust, but I'm glad I had a little civilian training before I started my training at my airline. It gives you a "recency of training", which to some airlines, US Airways specifically, shows you have a recent training course under you. Finally, it gives something to round out your résumé with.
When I went to do my ATP, my local education office told me that the GI Bill wouldn't cover just the ATP (btw, you're doing the right thing getting the written out fm the way with Sheppard Air), so I did the type and was fully reimbursed, minus hotel costs. Later, I found out they were wrong, but I was still glad I did the 737 type.
You should be able to check your flight records out from OSS, I was able to when I went to an interview at Eagle. They just asked me to return them when I was finished. After my fini flight (which was my final out date, which was the date of my retirement ceremony, which was the day before my effective retirement date, lol), they signed the folder over to me, I took that along with my Form 8s, and went off to my JetBlue interview. I didn't do a thing to my green folder, other than audit it when I transferred it to LogTen Pro. I never found errors before, but I found a whole bunch of 781s during copilot school that weren't in my records. It was only dual received, but gave me a little bump.