How do I go about traveling with extracted DNA?

The exact answer will depend on which countries you're travelling through - I'm going to presume you're flying domestically within the US (AK to somewhere in the lower 48) and then onto Austria directly...

First the Dry Ice :

There are 3 places that you will need to worry about - firstly security, then the relevant airline authorities, and finally the airline(s).

Within the US, TSA DOES allow dry ice in both carry-on and check-in luggage - up to the FAA mandated limits.

The US FAA also allows dry ice, up to 5.5 lbs (2.5 kg), with specific rules around how it is packaged.

Each airline will have their own requirements/regulation, so you'll need to check with whichever airline you are flying with to be sure. To pick one at random, United Airlines does allow dry ice, with the same 5.5 lbs limit the FAA allows, however with the added condition that it must be packed in a hard-sided ventilated container.

There are also generally restrictions on the maximum amount of dry ice that can be loaded onto a single aircraft, so most airlines will require you to declare that you have it, and give an approximate quantity when checking in.

As for the DNA samples themselves, some countries do have restrictions around importing organic materials. I would suggest contacting customs in Austria (and any other countries you are transiting through) to confirm that there is no additional paperwork required.