Both batteries will be charged at the same time whether using the alternator or charger. The alternator should charge the batteries quicker than the charger. The battery that is the "least" discharged will charge to "Full" quicker. The older a battery is, or the more that it is used, the less of a charge it will be able to hold, which explains why your battery runs down quicker. To test the batteries, you should get a battery load tester, which can be obtained at most auto parts stores that sell batteries. It not only tells you the voltage of the battery, but also how much of a load is remaining on the battery. It can also tell you if the alternator/charger is working. I just bought one and used it on my fully charged starter and house batteries. The starter batter looks good, but the house battery is starting to get weak, which is expected.
The factory installs Exide type 24 combination starter/deep cycle batteries on the R21 EC. These are 70 Ah batteries - which is not very much. The combination batteries do not last as long as plain deep cycle batteries. The plain deep cycle batteries can handle more charging cycles than the combination batteries.
The first night I anchored overnight, I anchored about 10 pm and ran the refer all night, and the house battery was able to support it. On my next cruise, I discovered my Garmin 5212 has an Anchor Watch. So I ran the Garmin Anchor Watch and the refer starting about 5 pm, and about 3 am in the morning, the Garmin started beeping to let me know the battery was low. I have another bank of house batteries which are Type 24 12 volt 70 Ah batteries connected in parallel, which gives me 140 Ah, which is not enough to run my inverter and other DC items. I just removed those batteries and installed two Interstate GC2 6 volt 232 Ah batteries, installed in series to give me 12 volts. These batteries are great. They run my inverter/microwave. I am going to move the refer and Garmin over to these batteries, and leave my original house battery for the lights, pumps and radio. The Interstate batteries are deep discharge batteries and should last a lot longer than the combination batteries I was using. The GC2 are golf cart batteries, but Interstate also lists them as RV/marine batteries. Batteries need to be listed as "marine" batteries so they will hold up to all the pounding they can get on a boat.