Good character choices would be those with high hp such as Flying Castle, Marie Poppo, or those with passive ability to heal themselves such as Shifu Robot and Suguri (Ver.2). Larger boards such as Sealed Archive or White Winter are best for avoiding pursuers and battle tiles, and enabling the Regeneration field effect can help a bit. Having plenty of heal cards like Saki's Cookie, Dinner, or Pudding can certainly come in handy, as well as aiming for Home panels for the extra heal.
Another viable strategy is circling your home on the Space Wanderer board, getting stars and guaranteed heals every other turn, albeit with the Random Warp field effect switched off. As long as you can avoid battle, you're generally set.
The Treasure Island (Night) board, introduced Summer of 2017, is perhaps the best choice, as the way the warp-move panels are set up guarantee you heals every single turn.