I would suggest walking with flat feet rather than the usual rolling foot technique most people walk with. This means that rather than landing with your heel 1st and rolling onto the ball of your foot, you should walk in such a way that your whole foot contacts the ground at the same time. This should keep you upright on flat ground. You will also notice that walking this way you will start to use your hamstrings and glutes to walk rather than your calves.
If you're walking uphill, I would suggest step kicking, like that used by mountaineers walking uphill in snow or ice. From your description I get the impression that the ground has thick mud rather than a thin layer of mud / moss over slippery rocks. To walk with step kicks, on every step you should kick the ground with the toecap on your boots. This will push your foot through the soft mud and essentially create a small flat 'step' which you can then use to walk on rather than walking on an incline.
Walking poles might also help you with stability as you can drive these into the ground and use these to take some of your weight, so you put less weight through your feet.
Amalgamating the alast 3 paragraphs from the answer from @JamesJenkins into this as per the request of the OP.
You need to increase your traction, and you probably want to do it
without changing your boots. > Strap on traction devices.
There is a huge selection of Ice Cleats these tend to be very short to
about a half inch. Longer spikes are available in Lawn Aerator
Sandals. The longer the cleats/spikes and the further they are spread
apart the more difficult it will be for them to fill with mud.
Allowing you to get consistent traction. But at the same time the
harder it will be to balance on them while walking. Which is the
lessor of two evils, the mud or the spikes?
In reality you should probably just spread crushed gravel (like 3/4
minus) along the path. Have a load of gravel delivered at the start of
the trail, take a wheelbarrow and shovel, spread gravel ahead of you
along the path as you walk up it.