Jack,
Using the ON/OFF switch that should be located directly on your Cobra 400 watt modified sine wave inverter, verify the AM radio squeal interference is really coming from this inverter. If it is, then I would recommend that this inverter be replaced with a pure sine wave type as I am sure that will stop your interference problem.
I guess I don't know what equipment in your Born Free your inverter is setup to power. In may case I installed a 300 watt Samlex Model PST-30S-12A pure sine wave inverter in the cab over entertainment center to power a 45 watt Samsung 20" LCD TV and a 25 watt VCR/DVD player. These two items replaced the original 90 watt 20" Panasonic CRT TV/VCR combo that failed.
I would generally recommend that your inverter's wattage rating be approximately twice the sum of the wattage ratings of all the equipment you intend to power with it to make sure the inverter will handle the initial heavy startup loads some equipment draw.
My inverter was plugged into the 12 VDC cigarette lighter socket that was located next to the 120 VAC power socket for the old TV in the entertainment center. Therefore, I power the new TV and VCR/DVD units all the time via this inverter even though they could be powered from the 120 VAC socket when AC power is available from shore line or the generator. I found out it is just a lot simpler to let the inverter power these two units at all times whether plugged into shore line power or whether the generator is running.
If of interest, I purchased my 300 watt Samlex inverter for $169 from DonRowe.com available at the following website link:
http://www.donrowe.com/inverters/puresine_300.html
If a 300 watt inverter is not enough power for your purposes, Samlex also makes a 600 watt unit that can be purchased from DonRowe.com for $246 and available at the following website link:
http://www.donrowe.com/inverters/puresine_600.html However, an inverter of this power rating should be wired directly to your coach batteries with heavy cables and not connected to 12 VDC power via a Born Free cigarette lighter socket which is fused for only 15 amps (and that is for all such sockets in the coach which are wired in parallel and connected to that single 15 amp fuse). Remember that a 600 watt inverter fully loaded will draw more than 50 amps from its source of 12 VDC power.
My 300 watt inverter installation is only called on to supply 70 watts, so it draws only approximately 6 amps from the 12 VDC cigarette lighter socket, well within the 15 amp fused limit. Born Free warns against using those sockets for powering inverters but at this low current draw, using it for that purpose has not been a problem for me.
Also if you want more information about the advantages of pure sine wave type inverters over the modified sine way type (and other info), here is a link to a FAQ on this subject:
http://www.donrowe.com/inverters/inverter_faq.html