[pushkins]

The "sinking" roof is a problem and should be addressed sooner rather than later as you mentioned.
What is the grade of this roof ?....1:12, 2:12, 3:12 ?

Shingles shouldn't be used on any roof under 3:12 (some shingle manufacturers will say down to 2:12, but you need two layers of roofing felt and most building inspectors frown upon 2:12 shingle installation.
This leaves you with the options of rolled roofing or steel roofing. Rolled is basically the same as shingles just in 3' wide sections rolled out the full length of your roof and nailed down just like shingles. Rolled will be your cheapest option.
You could always install steel roofing yourself but depending on how the flat roof attaches to the hipped roof, flashing it correctly might become tricky and much more time consuming.