Short Answer: Make use of Submarines to control the sea. get Aircraft, especially the B17, to hit cities. Use melee naval units to take cities.
Maximizing Your Strengths
As Washington you will get access to the B17 with the Radar technology. It is slightly stronger than a regular Bomber and starts with the Evasion and Siege promotions. This will make it do more damage against cities, and take less damage from interception. As you are playing Archipelago, most cities will have coastal access. After hitting a city with your aircraft, you can swoop in with a melee ship to capture a low health city.
Regarding promotions, try to build your aircraft in a city that has Barracks, Armory, and Military Academy, and rebase them after. The XP bonuses will let your aircraft start with two additional promotions. I strongly recommend Siege II, and Repair, which it unlocks. This will allow your bombers to attack every turn and heal after doing so. Another good option is likely Siege II, then Logistics, for multiple attacks per turn. This could be helpful for suddenly striking a target multiple times.
If you don't have one already, it may be worth establishing a forward city, to base your aircraft for bombing. The B17 has a range of 10 tiles, but this is expanded to 12 with the Range promotion, which will be available immediately due to having the Siege promotion. Taking range does delay Logistics or Repair. The alternative is to use aircraft carriers, which are more mobile but weaker than a city. Do not let a carrier holding multiple bombers be destroyed.
Also note that the B17 requires oil, so you may need to strategically take territory if you lack this resource. I would prefer taking land-based oil, as offshore platforms can be pillaged by naval units.
I would prioritize cities which are more distant from his cities, or closer to yours. You want to have an easier time bringing reinforcements. Remember you can conquer the city-states, some of which should be far from your opponent.
When at war, city-states seem to favor using land units. Guard your borders with naval units if their units are an issue. Embarked units are much easier to destroy.
As you don't have Radar yet I would hold position until you do. On archipelago this means controlling the ocean. You should focus on technologies that improve your naval presence, or bring you closer to bombers.
Refrigeration - This gives access to Submarines, which are invisible except to other submarines and destroyers, which require additional research. Although they cannot take cities, they work great for raiding trade routes due to their invisibility. You can also pillage work boat improvements to hamper your opponent.
Radar - This gives you the B17 which I discuss above.
Combustion - This gives the Destroyer, which is the best answer to any rival submarines. It is also a melee ship so it can take cities.
For domination only, I would max out the Honor Tree. It has many good bonuses, and gives gold for killing units when completed.
The next best option is likely to take good tenets from an ideology (see below) if enabled.
Maritime Infrastructure from Exploration gives a good bonus to naval cities.
If you are playing with Religion, consider finishing the Piety tree. It gives a Reformation Belief, and Religious Fervor and to a lesser extent To the Glory of God give good uses of Faith into the late game. Don't bother with this tree if your religion is not dominant within your own cities.
Take a good look at your Ideology (If you don't have one, get into the modern era, or build more factories), and look for the most helpful policies.
For only domination I would favor Autocracy > Order > Freedom
Regardless of the ideology you picked, it unlocks a wonder that gives an additional social policy, as well as some other bonus.
Also pick tenets that benefit domination, don't take anything for Tourism or Spaceship Parts. Take happiness tenets if you need them.
Autocracy: Elite Forces gives a boost to wounded units. Most of the second tier policies are good, and depend on what you need. Nationalism, Third Alternative and Total War all benefit building a large army. Lightning Warfare is not great on Archipelago, where you need more focus on naval and aircraft. Level 3 Clausewitz's Legacy should only be taken if you plan on engaging in an extended war, don't waste it if there is peace.
Order: Less direct combat benefits, more on building a strong empire. Again level 2 has the best options. Five-Year Plan and Party Leadership benefit all of your cities. Only take Resettlement if plan on founding (and holding) enough new cities. Level 1 Patriotic War does have some synergy with the American ability to buy tiles.
Freedom: If you have the Freedom Ideology, I would honestly take a look for the regular Social Policies. Regardless, Their Finest Hour gives a decent boost to cities, while Arsenal of Democracy and Volunteer Army help your offense.