Sounds like your tank hadn't even started cycling when you put the fish it - but since you didn't have a test kit who knows? You might have actually been close to having completely cycled - it's just strange that you have ammonia left over when instead you should have had only nitrates left over (then you would have cleaned your tank and added your fish and all would be well).
The care methods that have been out there have always been a bit spotty. When I had a tank about 30 years ago - the store just said to add a some chorine remover and wait one day before adding fish - and everything went just fine for me. I am getting ready to purchase a small Nano aquarium to put some extra fish in - the instructions on their box also says to put in something like prime then wait a day and put in your fish. So nothing has changed' and that seems to have worked for the customers of this company because you don't hear any thing in the review of the tank that mentions cycling their tank or if the box instructions to just put in fish after 1 day killed any of them. In fact - all the reviews are really good. So - since only the ammonia in your tank is high (and not very) maybe you should just leave it alone and wait for your filter to start cycling the tank.
I have a similar problem that's been a major pain. My tank had been cycled (used ammonia - no fish) for couple of weeks now, but at some point in that process I did mess with the PH because our tap water Ph is in the 9.0'sot 10's. . Looking at my notes I did get it down to 7 BEFORE I added any fish Got it in the 7's before adding fish so no big deal other than when I added three species of fish (Zebra Danio's, Cherry Barbs and Cory Cat's) I lost a newly purchased Zebra Danio overnight - which I've heard isn't unusual - some fish can't take any stress. Life went on for a week or two so ordered 2 Blue Powdered Gomorii's, and 2 Red and Blue Gomorii's. Suddenly I had a little ammonia Spike to .25 and then to .50 then back to .25 (unfortunately I had already ordered the fish) - Then, following a fairly aggressive cleaning and water change - I lost 2 Zebra Danio's on the same day - took some readings found quite a jump in everything Ammonia was .50, Nitrates 1.0 and Nitrates to 2.0. I thought maybe the dead/briefly rotting Danio's caused it. Did a water change again). After the water change the Nitrates actually went UP and the ammonia and nitrites have been stayed at .25 and 1.0 respectively. Right in the middle of this mess I received the fish had ordered 2 more Gouramii's, and 5 Zebra Danio's to replace the ones I had lost. All this time I'd been using Stability and other solutions which some forum members disagreed with so I had stopped that All the new and old fish have been strong and healthy thankfully. I really feel like I should try a third water change but I don't want to stress the new fish - so I'm waiting a few days or so before I do this. The PH briefly jumped up to 8.8 but then settled down to 7.4 where I'm leaving it.
I think I'm also going to pick up that little 5 gallon tank and move the Zebra Danio's in there - It is starting to look overcrowded because they school CONSTANTLY and while I think it's funny - the other fist all look like they are developing a plan for murder. While 5 gal is small - it's about how much room they have to school in the current tank and they won't have all the other fish in the way. The fish in the old tank should be calmer without all that commotion. Of course I'm going to cycle that tank as well. (this is the tank I mentioned earlier with instructions that just say to use a solution to de-clorinate, let sit overnight, and add the fish)
In the meantime I am still scratching my head about why suddenly it looks like my tank is mid-cycle instead of done with all of that business - what did I do? I left some plain tap water out last night and I'll test the tap water this morning and see what we normally read - even though I kind of did this when I first got my kit and - other than PH which was too high to read at 9.9 - everything else was all zero's). Then at some point this weekend I'll try to do another water change on my bigger tank and hope I don't kill any more fish and the nitrates don't rise even higher.
So keep us posted on what you finally decide to do - and how it works. Thanks - sorry for my long sad saga.
Oh - one thing I did want to mention is that I have read that many CoryCats (and I love those poor panda's that you lost) are VERY sensitive to salt.- In your situation those have been the fish that have been dying. I know I've read in some books that you should add a little salt to your aquarium for reasons I've forgotten - but if you did that at some point that may be responsible for their deaths. Just a wild thought.