Can I put new counter over what's on my counters now or do I need to

what is the existing counter top and what are you thinking of for the new one

Without knowing specifics like what type of counter you have, my best guess would be no. You are going to raise the counter height. Your new countertop may be affected by the problems with your old counter top. Sealing around the sink will be problematic unless you are going to redo it. So I would say I think it's a bad idea.

Is there some reason you don't want to remove the current counter? You didn't mention the configuration of the old countertop so no one could give a reasonable answer to your question. Cannot imagine a new one installed solidly over an old configuration.

Since you will likely have a new counter professionally installed you should be talking with the installer. Chances are they will remove that old counter for you at little or no additional expense.

Best to remove existing counter before installing new.

It's best to remove the old counter first because if you put a new one on top, it's going to affect the sink and the plumbing. The new top will be higher and stick up over the sink. The only other thing would be if you're going to paint them. That wouldn't leave such a depth change. If you still want to try in, you'll have to take out the sink and disconnect the plumbing, plus even then you'd have to make sure that the old counter has no holes and is perfectly attached. I personally think you'll end up with a big mess. It would be a much cleaner job to remove the old one, make sure everything is good under there then install the new one. As Ken said, if you have a professional do it, they are going to remove the old counter first, probably at no extra charge and as Naomie said it's best to remove the old one first.

White vinegar with detergent and hot water. You might need to repeat a few times.