Based on what you have described, I am fairly certain that your holding tanks have emptied properly and are not the cause of your sink problems. RV's are fairly notorious for giving false holding tank level readings. The probe sensors in the holding tanks are probably dirty. You can use some Holding Tank Probe Cleaner to get the gunk off of the probes and they will hopefully start giving accurate readings again.
Just like the sinks in your house the sinks in your RV can get clogged. In all likelihood the P-Trap in your sink is clogged with grease, hair, etc. and not allowing the water to drain into the holding tank. You have several options to clear out the P-Trap on your sink.
1. You could try to use a bathroom plunger to see if you can move the clog out of the P-Trap and into the holding tank.
2. If the plunger does not work you can use some drain cleaner that is safe for plastic pipes. You should use some Eco-Friendly type of drain cleaner as you will end up dumping it into the sewer system.
3. If all else fails you will end up having to disconnect the P-Trap from your sink and cleaning it out manually. This job is actually very simple, but it can be messy. You are going to need a bucket to catch the water when you remove the P-Trap. You should also wear some rubber gloves when doing this especially if you have used drain cleaner in the sink.
Below is a short video that shows you how to perform this job. Even though the video is for a house sink, all the steps are still the same.
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