Lots of options in Los Padres...
In the Big Sur area, on the coastal side, you've got trails like the Pine Ridge Trail, the trail(s) out of Botchers Gap, trails from Salmon Creek Station, trails from Kirk Creek camp, etc. All should have water and be nice in the spring. All will also have poison oak and ticks though, so be careful with the dog.
You can also hike into the Big Sur mtns from the inland side. Trails leave from around Los Padres dam (Carmel River), Arroyo Seco camp, etc accessed from Hwy 101.
For Big Sur area ideas, check out the website: www.ventanawild.org They've got whole sections of their website dedicated to trails, trail conditions, trip reports, etc.
Farther south into Los Padres, you could leave from Nira Camp behind Santa Ynez and hike up, or down, Manzana Creek to the Sisquoc River in the San Rafael Wilderness. Check out www.santabarbarahikes.com for ideas around here.
Yet farther south, you've got the Sespe Wilderness and could join Jack's group for the BPL hike (nice area).
Carrizo is a neat place with a lot to see, but it's getting hot out there and it's already super dry out there this year. Probably not a good environment for hiking with a dog right now.
Heading inland, you've got the Golden Trout Wilderness entering from the west side of the Sierra (south of SEKI) off Hwy 190.
There's probably other parts of Sequoia NF (not National Park) that are also open for hiking now in the 5-8,000' elevation range that shouldn't have much, if any snow left and would be nice right now.