It looks like you have a bit of work ahead of you!
You got a pretty good deal on that Safari. However, it's going to take a bit of work. But there are lots of us that have been down that path!
First, it really appears like you probably have a bad axle. The axles used at the era of your trailer used rubber to absorb road forces. As it gets older, the rubber stiffens and stops providing a cushion. As a result, the trailer rides very bouncy, it causes twists (stressing rivets) and the bottom can things with the kind of results you photographed. This is a significant repair. Up from that, you could mail order one (there are a number of manufacturers of trailer axles with equivalent products). Depending on your skills, you could either install it yourself or hire a local mechanic to install it (they can also measure for ordering it).
The second thing I see is that you do have a bit of surface rust on your chasis. I'd suggest treating it before it gets worse. It is a fairly easy DIY project to treat the exposed metal (there are spray-on auto body treatments for rust conversion and paint that work effectively in situations like this).
There also seem to be a dark spot on the underside of the decking that might indicate rotted wood. Since your belly pan is open, take the opportunity to probe it aggressively. If it is compromised--you're going to have to replace it. You also need to check other places for similar damage.
Finally, there is the belly pan. I can't tell, but it looks like it was patched once before. On all the early 60s trailers I've seen, the belly pan is constructed from two long sheets of 4' wide aluminum sheet goods running the length of the trailer, joined at the center. With the monocoque design, the belly pan is an integral part of the stability of the whole upper body and is attached with buck rivets up in the walls. Consequently, you shouldn't see any lateral line of pop rivets underneath and you should see a center line seam where the two sheets of aluminum are overlapped and riveted together.
You may be able to simply re-patch or you may find you will need to get into more structural concerns. Replacement aluminum is not horribly expensive. However, installing a belly pan is very fiddly, tough to reach and complicated to install, since a full replacement requires removing the furniture and the inner wall panels of the trailer. That is time consuming, at the very least.
You are going to have to have someone experienced in this make an on-site analysis of whether the old patch is problematic or not in order to fully guage the scope of the repair. I don't know how the vintage community is in your area, but it's a good idea to find an experienced restorer (with structural experience with early 60s/late 50s trailers) to help you guage your situation. Checking the Vintage Airstream Club or members of this forum to find someone reasonably local. The advice you would get would be most impartial and probably will save you considerable money.
However, this opening does open opportunities--to identify if there are other concerns you need to attend to and for possibly installing a grey water system.
We've faced these problems ourselves, so don't feel you are alone. Hang in there--the result will be worth it!