I borrowed a Bissel and a Hoover hands down Hoover was best so I bought this one
Hoover MaxExtract Dual V Carpet Cleaner, Black, F7412900

it is an awesome machine. Cleans so well, better than the commercial one's we have rented, and dries very fast.
This model has the separate rinse water container, the detergent goes in it's own container and it mixes it as you use it, this allows you to do a separate rinse with just water to get out as much of the soap as possible.
I do a soap/clean stroke up and back then switch to rinse and do up and back then release trigger and do a dry pass up and back. Works great.
You do need to use some of the cleaner detergents, I also have asthma and scent allergies but these have not bothered me at all and I think the fact that I can do the clear water rinse helps because that takes out all the detergent and the scent.