Hey, welcome to the forum. I'm personally a fan of Subarus, don't know why just really enjoy them. Had a Forester for almost a decade and recently replaced it with a new Forester. Anyways, here are my thoughts on the matter, but please keep in mind I'm FAR from being an expert so please wait for the opinions of those more knowledgeable then myself.
The non-turbo Subaru engines are considered to be rather easy on oil, in my opinion using synthetic oil with 3k mile OCI (oil change interval) is wasteful. If it gives you confidence and lets you sleep better then obviously it's worth it, but to me I'd push my OCI out longer. For example, in my Forester I'm using dino oil and a 4k mile OCI to maintain the warranty. If you're going to stick to a 3k mile OCI then I think you should consider saving some cash and using a dino oil.
Check out this forum and read as much as possible, you'll quickly see Pennzoil Platinum (PP, Pennzoil's synthetic oil) is very highly regarded on here. Yes you could move "up" to oils such as Amsoil or Redline, but I personally think that's very wasteful for a 3k mile OCI. But I don't think the oil is at fault here.
You mentioned "I'm still going in every 3000 miles for a change and my oil isn't lasting that long (rough starts, poor performance, lost oil). One time between changes I ran completely dry with no signs of an oil leak." Honestly, any decent dino oil should be able to go 3k miles in a normal car engine, so if you're consistently having problems with a quality synthetic like PP, then you need to consider that it might not be the oil.
Now when you say you ran the car dry, I assume you mean it was just really low, maybe not showing up on the dip stick, but I doubt it went dry-dry. I've heard mix things, but generally a car does not last long after that, usually stories of this involve the car completely dying. But even if it was just getting really low, that's excessive oil consumption unless it takes you a very long time to rack up 3k miles.
I think you need to address this issue, I personally think it's more important than finding a new oil. If it was my car, the very first thing I would while waiting for other people to post their recommendations is I'd pick up a quart of oil and leave it in my car. Then every time (or every other time) I filled up with gas I'd also check my oil level to make sure it's not getting too low, and I'd top it up as needed.
Also, you might want to look to see if you're losing any coolant, and similarly look in the coolant reservor to see if there's any oil mixing in there. Subaru's in the late '90s and early '00 had some headgasket problems. While this was supposedly fixed by the time the '04s came around, it doesn't mean it isn't possible.
I'm not knowledgeable enough to help you diagnose the issue more then that, so please wait for people more experienced then myself to chime in. Though I will make 2 other quick points. First, I know people who have had great results using high mileage (HM) oils, helped address burning oil and other issues they were having. I personally have never used HM oil, so I personally don't feel that I could comment on it; but I wanted to pass that info long.
Also, on here there are a lot of people who love Auto-RX, I haven't heard anyone say anything bad about it other then saying that they personally don't believe its necessary. I decided to give it a shot and put some in an old beater SUV that I have, however it's way too soon for me to make any personal comments on this product.
Anyways, best of luck getting this issue addressed, honestly I don't believe it's the type of oil you're using (PP is really good stuff), and so I don't think switching oils to something "better" is going to solve your problems. Well, good luck