Outside of the cities and small towns the traffic won't be any different than any other time of year. Just remember the biggest vehicle has the right of way. The traffic police will probably be out so watch your speed. Get an early start. You don't want to drive after dark. Good luck and good travels
Drive during the daylight hours only. Expect large potholes, bumps and sometimes part of the road to be missing. Don't speed, keep lots of distance between you and the car in front. Expect the taillights, brakes lights and turn signals on other vehicles not to work. Check your mirrors often, other vehicles appear out of nowhere. For every vehicle you see, expect them to do the craziest thing you can imagine at anytime. Expect to be passed on the left or right. A two lane road is often be used as a three or four lane road. The centre lane markings are not respected, vehicles pass each other on curves and climbing hills where the view of oncoming traffic is obscured.
Expect people, animals and debris to be anywhere on or near the roadway. Often debris falls off trucks or other poorly loaded vehicles. Know where you are going and how to get there before you leave. Signage is often sparse and confusing.
Take your time. Force yourself to stop every few hours to stretch your legs and clear your mind. Try to avoid grinding your teeth as your jaw will get sore. Passengers prone to making use of the imaginary brake pedal or uttering shrieks of panic that could startle the driver should sit in the back seat.
Your travel dates are probably as good as any others.
Regardless of what others do, stick to conventional driving rules and keep cool. Keep within the 100 km driving speed, use your turn signal all the time. You may have to pass on the right, this may be unavoidable.
Consider using Waze. The Spanish language version has more details. If you are traveling with another person, use both -- one in Spanish, one in English. It is best to take the Santiago Circunvalaci?n. Unless you want to stop in Santiago to visit the Centro Le?n, the Santiago cultural center that gives a good review of Dominican culture in general. You could snack at their cafeteria.
Another option is to head for Bonao and have a meal at the Tipico Bonao (there are two -- you want the one to the south -- it shows as Av. Libertad), you have to take a right turn, but it is indicated. They have good Dominican food. They are usually full but service is fast. If they are full, register at the cashier and they will shortly get you a table.
Waze will guide you to pass through Santo Domingo. There are many routes, but you need to go East.
You also could make a stopover in Santo Domingo, to tour the Colonial City. It is on the way. You could have lunch at Buche de Perico. Then it is back on the Las Americas Expressway all the way to the East. If you want to stop midway, there is Meson Espa?ol in Juan Dolio for delicious seafood fideua.
You also could make a stopover in La Romana -- but if you want to continue, make sure to keep in direction of Higuey and not Casa de Campo. If you have indicated Punta Cana on Waze they will guide you to another express toll road and keep you from entering La Romana town.
If you need a break on the highway, there is Shish Kabab in a gas station right on the highway. It is a good place to stop. Then it is all the way to Punta Cana.
Take note that while there is lots going on from Puerto Plata to Santo Domingo -- lots of attractive goods to buy on the highway -- there is practically nothing from La Romana to Punta Cana -- even the scenery is boring. That is why it is good to make the stop over in Shish Kabab (they are famous for their "quipes" and Arab food Dominican-style) to keep your driving relaxed.
Puerto Plata to Nagua and then that 2 hour drive to Santo Domingo on the tolls road is a good option. It depends on where you want to stop.
If you decide to take that road strictly stick to the speed limitations of 50 km in the Los Haitises (very curvy area) and then 80 km for the rest of the road. The drivers will speed by you, but you will enjoy the scenery and will be safe. It is a lovely road.
If you need to stop, recommend the Texaco gas station. That road will take you to the outside of Santo Domingo, near the las Americas International Airport, and right on Las Americas Expressway.
You have been given a lot of good advice.
The only things I would add is to be extra careful of the guys on motorcycles. 99% (probably being kind) drive like drunk idiots, even though they may have their family on board.
There is a scam which seems to be more popular with the tigres on the north coast where they either tell you a part fell off or sometimes throw something at the car and tell you they have a shop near by and will fix it cheap. They drive along side you and point at the underside of the car. I suggest you keep driving to the closest gas station or plaza once they are out of sight.