No, it's not normal. The drip pan should be draining to somewhere else. You need to get to where the unit is and blow the clog out of the drain line coming off the unit itself. I usually use a can of compressed air for that, or else just take deep breath and give it a big HUFF to push the clog through.
Yeah, it sounds gross but whatever. It gets the job done.
If you can't do that you can try running a Zip-It or similar down into that drain line. Get the longest one you can find because sometimes those drain lines can be kind of long, but if you're lucky the clog is in the trap fairly close to the HVAC unit itself.
I hope you're saying that the unit is overflowing and the drip coming from it is landing in the tub, and not that the unit's drain line is routed so that it flows into the tub, because THAT would be, like, super janky installation work and whoever did it needs his butt kicked really hard.
Good luck with it.