Thea, if this were my plant, I would immediately remove it from its pot and check the root ball. Your plant may have no roots at all. The base of your plant, and its tips, should be firm, never "squishy". You'll have to post what you find if/when you examine that root ball. Is this plant leafed out or does it only have small, terminal leaves?
When it comes to sun, plumeria are sun-lovers, though during the extreme of summer here (mid-July), I might have to lightly shade a few plants that have a bit more sensitive leaves. The combination of good roots, lots of healthy, large green leaves, sun, and air movement, allows plumeria to transpire well. Thus, these plants are able to pretty quickly use whatever water they get. Be sure your plants are in well-draining soil with good drain holes, and don't allow them to sit in water-filled saucers. They don't like "wet-feet".
