RNA can be stable if it's dry but DNA is much more stable. I can't see DNA surviving undamaged on the surface of an asteroid, but I could see: DNA getting somehow embedded in a rock agglomerate, then the rock gets ejected from a planet, lands on another planet whole, and then the rock breaks and now you have dna on another planet. With enough deep time, this could occur (possibly) but I can't see it as an effective method.

instead, better to take tardigrades, induce tun state, embed that in rock, and let that float around for a while. They have pretty much everything needed to survive including safe storage for DNA, DNA repair enzymes, redundant DNA, radiation-resistant exterior, etc.