I'm not familiar with your climate, so I'll leave out mulch advice!
You may be interested in checking out 'hugelkultur', where wood is buried under areas to be planted.
I plan to get hold of some decomposing wood this year, as apparently it has all sorts of great effects, not least dramatically improving water retention.
All sounds a bit counterintuitive with all that buried carbon, and (they say) the first year can be so-so, but after that...

I'd use them after running them through the chipper.

Chip them and use them as mulch. I have not seen any increase in termite activity where I have used wood chips, from fresh wood or old, dry, wood in the 40 years I've been here.

Thanks for the advice. I'm going to chip it all and try it as mulch.

Branches and prunings make wonderful mulch, especially is mixed with other things like grass clippings, leaves, other crop residues. If it attracts termites they will help break down the woody residues and no harm will come unless you have a wooden building next to it. I would also suggest a high nitrogen organic fertilizer like cottonseed meal or alfalfa meal spread on the ground prior to applying the mulch.

Here is a link that might be useful: In the Charamon Garden

I have been reading about arid land that is being replanted with trees, and also about the education of local rural people about using mulch to preserve moisture and help fertilize plants. They show how to use small branches stacked around plants, both live and dead. Tree and bush trimmings can be cut up and placed at the base of plants. I have been doing that at my house for a few months. Huge trees (Jacaranda) are always littering the yard with dead twigs. I break them up or cut them up and stack them artistically as mulch, very deeply. It looks pretty good! The loquat trees I did this to are already throwing out a ton of new leaves. Some leaves were also mixed in to the branches or thrown on top.