If you have thoughts that are repeating in a loop, most likely you are focusing on them or obsessing over them, intentionally or unintentionally, rather than "not resisting" them.
My understanding of not resisting thoughts is this. If you are practising mindfulness on the breath, then when any thoughts appear, just take note of the fact that the thought appeared, then go back to the breath. This way, you are neither focusing on nor suppressing those thoughts. You are instead, focusing on the breath. As soon as you realize that you left the breath to think of something else, your focus returns to the breath.
If someone tells you "don't think about monkeys", what would appear in your mind when you focus on that? Monkeys. The best way to not think of monkeys is to focus on something else. In this case, we are focusing on the breath and eventually, the mind will stick to the breath and the repetitive thoughts would rarely or never arise.
If you face problems, then you may want to solve them or mitigate their effects, if that's possible. Repetitively thinking about the problem does not actually solve the problem. Instead, it results in depression. Solving a problem and obsessing over a problem are two very separate things.
In the solution process, you may list out actions to take, execute those actions at the appropriate times, then review the outcome of those actions at the appropriate times. In between, while waiting for the "appropriate times" to come, if the repetitive thoughts return, then mindfulness meditation would be helpful to bring peace of mind.
Thanissaro Bhikku instructs in his essay:
If your mind wanders off, gently bring it right back. If it wanders
off ten times, a hundred times, bring it back ten times, a hundred
times. Don't give in. This quality is called ardency. In other words,
as soon as you realize that the mind has slipped away, you bring it
right back. You don't spend time aimlessly sniffing at the flowers,
looking at the sky, or listening to the birds. You've got work to do:
work in learning how to breathe comfortably, how to let the mind
settle down in a good space here in the present moment.
This Scientific American article tells us the benefits of mindfulness for depression and anxiety:
As a remedy for depression and anxiety, mindfulness meditation may
help patients let go of negative thoughts instead of obsessing over
them. Training people to experience the present, rather than reviewing
the past or contemplating the future, may help keep the mind out of a
depressive or anxious loop.
Basic Breath Meditation Instructions and A Guided Meditation are two essays by Thanissaro Bhikku that instructs one into breath-based mindfulness. If the Five Hindrances disturb you, then there are antidotes mentioned in this essay by Ajahn Brahmavamso. To make mindfulness easier, Thanissaro Bhikku suggests using the "Buddho" word in the second essay. You might find that very helpful.
