You're experience is typical of runners who first try swimming. Swimming uses muscles you haven't worked out or built up before. It is also a more complete work full body out. Also, you probably have low body fat so you get no help from buoyancy (same as regular swimmers, but they are used to it already).
Finally there is the whole addiction to oxygen thing.

The first thing I'd recommend is to stick with it, it will get easier after a week or so.

Second, you need to learn how to turn your head and breathe while doing free style - I assume you're doing free style because you are in a triathlon, and you want a better time. Practicing breathing is key...I'd suggest you just get good at it turning your head in one direction every time rather than learning to do it both left and right...that takes more time and practice when your starting from the beginning, and you don't really need to do it in both directions.
If you can't get the hang of it well enough, then you could always take a slower time and do side stroke where breathing isn't an issue.

You oxygen carrying capacity will also improve as you swim more making it easier yet.

I can't recommend swimming enough - great work out, low impact (no wear & tear on joints as you get older), great for getting a natural alignment of your vertebrae without traumatic manipulation.

The biggest downside is just finding access to an reasonably big sized pool where the chemicals are managed properly. If you can find an infinity pool somewhere, they seem to have a better filtration system which means less chemicals needed.
In the city, you will likely have to get up really early and wait for a lane. Making friends so that you can share a lane helps a lot.

The 2nd biggest downside is that your appetite will increase a lot - more than other kinds of work outs, and when you stop swimming, the appetite doesn't stop with it so be careful.

I had an injury that has kept me out of the pool this summer, and it is driving me crazy. You'll miss it as much as running if you get into it.

Hope this helps in some way.