maya__
As a gardener with several Oleander plants, I was surprised to learn that a fatal disease, OLS, is infecting plants in AZ & So. CA. The plants are so toxic, that I never considered any insect would be interested!
The suspected vector, the Glassy-winged sharpshooter, a leafhopper, is native to the SEUS and probably transmits other diseases in this area, but not OLS, so far!
Are you removing and destroying the plants, as soon as the disease is recognized?

Unless you have a source of cheap materials, wouldn't erecting trellises + plant material, be as expensive as planting shrubs?

Sorry, but I wouldn't recommend Wax=leaf Ligustrum or Honeysuckle, as both are invasive pests in this area and I spend many hours each year removing seedlings from the garden.
Why not a combination of plantings?
Some narrow trellises, 4-6' wide x 8' high. I would recommend an evergreen, woody vine, that I use and like very much: Trachleospermum jasminoides, known in the South as Confederate Jasmine and used as a groundcover or a climbing vine on trellises, arbors and pergolas. Blooms in early springtime here and is very pleasantly aromatic.
I acquired an attractive variegated leaf form last year and am testing it for hardiness in my Zone.

Some shrubs that I use for shielding the off-premise Uglies, are Camellia japonica & C. sasanqua, Pieris japonica (green and variegated foliage types), Aucuba japonica (Variegated Gold-Dust plant), Mahonia bealei & aquifolia, Osmanthus heterophyllus 'Variegata' & 'Goshiki' (both very slow growing), evergreen Rhododendron, native & Japanese Hollies (Ilex species & cultivars), Callistemon sp., Illicium floridanum & I. parviflorum.

To hide "bare legs", in cases where necessary, I have used Indian Hawthorn, Dwarf Loropetalum, 'Otto Luyken' Laurel, evergreen Azaleas, Carex and other evergreen perennials, plus dwarf varieties of Chamaecyparis pisifera and Yew.

Since there is concern about the disease affecting economic crops there, maybe there will be a cure for it in the near future. Let's hope so!