Reinforcement learning for continuous state and action space
As you say, policy gradient methods work better than value-based methods (like DQN) with continuous action spaces.

Your problem seems very similar to the MountainCarContinuous environment. If you check the leaderboard, you can see how the top score uses A2C. You can find a clear implementation of the same algorithm (using the same environment) in the following notebook: Continuous MountainCar Actor Critic Solution.ipynb. 

Here you can also find a collection of Policy Gradient Algorithms. For example, PPO has proven to be very successful in environments like the one that you describe. 

TL;DR Look for Policy gradient methods.