Mixing door is one area to look at...but
IF yu have a Cabin filter and it is completely clogged with soot and dirt...yu get a similar issue. They should be changed out every 15K or 1 year. Located behind the Pass. Glove box.
Check if yu have a cabin filter.. my 06 HL does and my 01 Avalon does too. My Avalon had a similar issue and replacing it fixed the air flow problem.
It is possible that yu don't have a cabin filter... then yu will need to focus on the
air mixer. Alot of tear down to the lower dash will be needed.
Good luck
LT