I don′ t think that House Hlaalu suffered that much, they should still have the king Hlaalu Helseth. Besides, they are most likely the strongest House at the moment, since House Telvanni was destroyed by the Argonians(damn worthless slaves), House Redoran and House Indoril were already at the edge during the reign of Uriel Septim VII, and House Dres had to abolish slavery and introduce another economic system.
Actually, does anyone know where Helseth even is at the moment? Last I knew the Argonians were still holding onto the southern half of mainland Morrowind, which would (or could; I haven't heard specifics) include the capital city of Mournhold. Helseth's either fled for higher ground or he's busy being a royal ransom. Edit: Although at the time Skyrim takes place, he's either very old or a bit dead. So maybe the better question is whether or not he has any heirs, and if they're still in power in what's left of Morrowind.
Speaking of Helseth, he actually helped Dres out, IIRC. He married a Dres noblewoman at some point after Tribunal. Even though he abolished slavery later, it was still while in an alliance with House Dres, and those kinds of connections with a ruling family would still go a long way to helping the rest of the House out. (Or at least those portions of House Dres which were on friendly terms with the new queen's family.) It's Houses Indoril and Redoran that really got the short end of the stick in the post-Nerevarine crisis years.