Let me start over in answering this. A converter takes 120 VAC and "converts" it to 13.0-14.5 VDC to bring the batteries to full charge and then maintain them at the preset level. The preset level really depends on whether it is a smart charger or a "dumb one". Dumb ones pretty much keep a constant voltage applied-- generally in the 13.2-13.5 VDC range. A smart charger will start out at higher voltage if the batteries are discharged and then taper off to a lower voltage after the batteries are fully charged.
So, a converter normally WILL run all the time-- supplying constant voltage. The amount of current (amps) it sends to the batteries depends on the draw-- once the batteries are charged, they will pretty much match their output to compensate for your electrical use.
To evaluate the converter, use a digital voltmeter (they start under $20 at any auto parts house, Radio Shack, etc). Tell us voltage at the batteries with converter off (or shore power and generator off) and then with it on.
And, as Rich said, if we know the brand and model we can give you more specific information.
Brett and Dianne Wolfe
2003 Alpine 38'
Moderator, FMCA Forums
Chairman, FMCA Technical Advisory Committee
Member, FMCA Long-Range and Development Committee 2007-2009 and 2014-
Moderator, http://www.dieselrvclub.org/(FMCA chapter)