Yes, Polynesians (and Micronesians) were aware of other islands and often traded with them. There were certainly battles between islands/groups of islands and centralised political control of large groups of islands. The Tu'i Tongan Empire is probably the most famous of these.

In the Marshalls, as in many other places, the chiefs, or Iroij, were locked in a perpetual battle for supremacy. It's documented that in the early 1820s, an Iroij, Lamari, of Aur atoll had succeeded in gaining control of the entire northern half of the Ratak chain and was in the middle of planning an assault on Majuro in the Rālik chain (Hezel 1983: 94).

The low-lying coral atolls around Yap paid tribute to the high island as insurance for times of strife like famines or natural disasters. In turn, Yap was in constant contact with Palau, with which they would trade for their Rai Stones.

There's been several threads on similar topics before (e.g. here), you should have a look for them.