How can I remove water vapor from an airstream without a cold trap?
You won't easily be able to use some other liquid to pull the water vapor out of the air. I think the cheapest option is a hydrophobic membrane or filter that you feel comfortable using. A commercial desiccant system will be the highest performance next to cryogenic cooling but may not be cost effective (depends on the scale you are working at). You can also try salt in place of silica beads for absorption drying versus adsorption drying; I imagine the silica will be more efficient but salt will be cheaper. When using things like silica, salt, or commercial desiccants you may run into issues of too much pressure drop as the gas travels through the packed column. You will have to figure out how often the desiccant media needs to be regenerated/replaced to make sure your drying is adequate.