Plumbago needs sun, but there are lots of others you could include:
Aconitum - blues (also some creamy yellow)
Pulmonaria - blues to pinks
Anemone sylvestris and divaricatus - white
Japanese anemone 'Whirlwind', 'Honorine Jobert'
most campanulas will tolerate at least part shade if not more
Iris cristata - blue and white (white is hard to find)
Iris japonica - pale blue
Cardiocrinum giganteum - giant lily, white
white astilbe
Corydalis 'Blue Panda', 'Gold Panda', purple leaf - blue
Dicentra eximia 'Aurora' - white
Dicentra spectabilis 'Alba'
Solomon's Seal
white epimediums
many hardy geraniums will tolerate part shade, including Rozanne, G. macrorrhizum 'Album'
Polemonium - blues

and for something larger, hydrangeas, Pieris japonica, azaleas (last two, whites only)

Blue lobelia, hydrangea arborescens (Annabelle is one of the cultivars, but the straight species is nice).

How about forget-me-nots? You can get them in both blue and white.

Sanguinaria canadensis and Parochetis communis which is frost tender but has strongly blue pea-shaped flowers over clover-type leaves. Alkanet can be obliging about flowering in light shade.

And the Tradescantias...

Plumbago will bloom in the shade, just not full shade. The plants will not be as lush, will bloom later, and will not bloom as profusely as they do in the sun, but they do bloom. The bright blue really adds to the fall shade garden.

Phlox divaricata comes in blue or white, and blooms well in pretty heavy shade in spring.

Torenia (a.k.a Wishbone Flower). The most reliable blue bloomer in my shady garden. Did very well all summer, with a max of 2 hours of filtered sun. I've seen in blue, violet-blue, and white. Moderate reseeding in my climate.

Small(ish) Clethras 'Hummingbird' or 'Sixteen candles' will be great structural addition to shade garden.

Grab a copy of "Made in the Shade' - cheap on eBay.

gardengal provided a great list; I highly recommend the Corydalis 'Blue Panda', it has a really long bloom time and is a beautiful light blue. Japanese anemone comes in many different sizes for whites and is a great late bloomer. Sweet woodruff, tierrella (foam flower), and rue anemone are great too. I planted plumbago last year in a pretty shaded area and it bloomed nicely.

The tall Lobelia comes in white as well as blue, thou the white is harder to find. Gentians often need shade and many come in blue.

Heavenly blue morning glory blooms for me in part shade, thou its cousin, white moonflower, wouldn't.

Springcherry

How about Tricyrtis aka toad lilly ? There are varieties that comes close to blue flowers and they are interesting foliage due to very different leaf shape , growth habit than other shade plants .

I am putting in several types of Campanulas, carpatica'Blue and White Chips'-they can handle part shade, Campanula persicifolia coerulea,two foot one that will bloom blue (light shade), the rotundifolia (which is called Scottish Bells-needs lt. shade-sun), and C. cochlierofolia Bavarian (sky blue-takes lt. shade to part).

The Blue and white chip variety bloom all summer into fall. The C. persicifolia coerulea bloom all summer, so I am adding Adendefolia (Ladys Bells) to extend the blue through fall.

Istanbuljoy

Browallia is said to take some shade.

Springcherry

Wood asters come in both a purplish blue and a white.

columbines come an a purple that looks like blue...they love the shade.

Jacobs ladder is blue.

Geranium 'Biokovo' blooms white with a pinkish tinge. I have two different areas of them -- one that gets quite a bit of afternoon sun, the other that gets practically none. The ones in the sunny spot are bigger and bloom more, but the others are doing just fine. Brunnera macrophylla blooms blue, but it's over with so early in the spring. Goat's beard (aruncus dioecus) blooms white, but later, and those can get pretty big, I've heard. Mine aren't so far -- about the size of a big astilbe.

Kristi,
You said the Torenia re-seeds moderately. Does it still form a ground cover? What cultivar do you have? I am planning a shade garden, but I'm sure it will be late August(Happy Birthday to me!) before it happens. I have been wondering what to use between the plants. The google pics I searched up looked quite promising, but they were not grown in Texas.

Omphalodes - blooms early, usually blue but available in white, and in shade. Ground cover.