120 F, roughly 50 Celsius, is not to be considered an elevated temperature. It's the "warm to the hand" temperature to which a welding end should be heated up when room temperature is cold.
Giovanni S. Crisi
Sao Paulo - Brazil

Unless the vessel also sees very low temperatures, and CVN testing is required, the group numbers don't matter with regards to procedure qualificaiton as a change in group numbers is only a supplementary essentail variable.  The 308L is a logical choice for joining to a higher grade stainless steel, as it will have adequate ferrite to avoid cracking issues, while will match the corrosion resistance of the 304L that you are attaching.  So if the 304L is adequate in its location, the 308L will be as well. 

As far as your temperature is concerned, welding at 120F is not a problem.  The base metal would be at that temperature after the first pass of a multi-pass weld anyway.  Just make sure the maximum interpass temperature does not exceed 350F.

You want to keep the stainless rod dry, but in order to avoid porosity, not hydrogen cracking as you would with low alloy steel rod.  So if you can weld without porosity, your storage conditions are adeqate.  You can't hurt the rod by putting it in an oven at 200-250F.  However, the best route is to consult the manufacturer's recommendations.

Thanks guys - great answers all around & a great site.

Best Regards