“Awash” and “brimming” may work. Though the Thesaurus lists them as synonyms of “full”, they don’t conjure up the “fullness” of “full”. In addition, “teeming” may work (a salute to the Teeming Millions!). Or maybe “overrun”?
What about “lacking”? It drums up images of low quantities, but there’s still some left (of whatever material you’re using). Or maybe “depleted”.
Anyway, I can’t think of an “official” word for “almost full/empty”… I mean, that’s why the word “almost” exists, isn’t it? If there is such a word, though, I’ll bet fifty SPOOFE points that it’s longer than three syllables.
Skosh is used for almost empty. From M-W.com
Main Entry: skosh
Etymology: Japanese sukoshi
: a small amount : BIT, SMIDGEN – used adverbially with a <just a skosh bit shook – Josiah Bunting>
Not what you’re looking for, but the empty space (or its volume) in a not-full container is called the “ullage”. Obviously this makes a lot more sense for an almost-full container.