Saturn mantra in graha sutra is amazing... I recommended to a person who got a db Saturn and Saturn Dasha running, they do it regularly... they are enjoying not doing anything and happily waiting for things to happen!!
It's in graha sutras under japa section. I have attached the screenshot here. It needs 23k recitations.
You see the English version for learning.
First, I believe you're going to achieve/receive what you have to from the Saturn houses anyway... to whatever degree your chart indicates.
Second, I don't think the "meh" attitude is actually truly letting go, or being detached as a true Saturn quality - which is a good attitude where needed. This true detachment is acquired by experience and contemplation, I would say. You still need to do (or pay your dues) the things from the Saturn house and so on, but being present in these situations is to me, the first step, because it will give the clarity you need in the long run. Simplicity is key when it comes to Saturn and its activities. I would not overthink this and get too confused.
Being in Rahu/Saturn has taught me a few things about simplicity and the need to actually put things in order. Seeing where I was tripping and causing me trouble, using the Venus and Mercury approaches to help out Saturn (I don't have that helping out my Saturn originally, which is actually pretty starved by the Moon) - and being BRUTALLY honest with myself, will help out perspective. Taking the junk out is really necessary. Simplify!
I believe working with discipline and not wasting time with stupid things is a good way to improve whatever outcome you're going to get. Focus on what is important, discard the garbage - be focused and use your energy wisely. You cannot control the outcome per se, but you can control how you use your precious time and all other resources. Not counting emergencies of course - so understanding that what you can do is in your hands, and the other part is up to God, or fate, or whatever, is always a good mindset to have. Keep yourself in order as much as you can - have a schedule, keep your home clean, your body, and everything else - it has worked wonders for me, and amazing clarity specially for my Saturn houses! And rest!
Oh, and acceptance. See the junk, take it out, accept things, move on. Be nice to yourself, take a nap.