It sounds like your blend door actuator. There is a difference that nobody talks about with the "temp air blend door actuator" and the "blend door actuator". The one you need is on the drivers side near the gas peddle and it doesn't look too hard to replace. Could probably be done by yourself if you can work in tight spaces.

A way to test it is to look under the dash by the gas peddle and you will see a little black box and some white gears behind it. Turn the control knob to switch back and forth and look to see if the gears are moving.

I am having trouble with my "temp air blend door actuator" and nobody seems to talk too much about this one.

You can find the part online. Hope this helps.

2007-2010 Hyundai Elantra Air Flap Actuator - Dorman 604-304 - - PartsGeek.com

Going in the right direction
 

You are very close at this but I'm going to say the issue is at the climate control , which provides power to signal the blend actuator to open or close,its an electrical issue at the climate control replace it or test it. Hope it helps

Have you done a flush of the cooling system and a flush of the heater core?
If the heater core is clogged then it would present a similar symptoms as low coolant level

I know this is old but did you ever get the problem resolved? I'm having the exact same issue in my '08 Elantra.

Blend door actuator

2003 Elantra heat problems
 

OK, so I have had my 2003 Elantra since 2011. It never has had heat. I tried flushing the heater core and not thinking it makes a difference, I hooked my water up to the connection closest to the drivers side, and water (a bit rusty) came out fine from the other port and I flushed it till clear, but still no heat. Fast forward 7 years till now and I decided to take a second look at the heater core because one person had said his hoses were switched and when he reversed them it fixed it, but many others said it made no difference.. (though I had looked at the inside controls etc without finding anything). So I went back and tried flushing through the port on the passengers side. NO FLOW EVEN WITH A LOT OF PRESSURE!!! So I went back and tried the other one again and sure enough, plenty of flow, and by this time I had determined by other means that the hose that had been hooked up to the passenger side port was the water input hose, so, now I reversed them (they had been like that since I bought the car used in 2011..and then I remembered that the person I had bought it from had replaced the water pump shortly before that. Could they have switched the heater hoses? Or more incredibly, they had always been that way?) Anyway, now I was hopeful, and sure enough!! Plenty of heat at 2000 RPM,, and some at less, but no doubt by now the core is less than perfect, but I had ZERO heat before! I was just about to buy another core thinking it was completely blocked, but what a surprise that would have been, had I done it without reversing the hoses. FYI, the upper outlet hose, most to the drivers side is the water outlet and it connects to the port closest to the drivers side. The return is from the port to the passengers side connecting to the motor lower and to the passengers side of the outlet hose.

If you want to replace the heater core then please search youtube
There is a way to do it without taking out the whole dash.
The heater core has extended metal tube going to the core.
Cutting out the old heater near the core lets one remove the old core without removing the dash
And when installing the new core (cut near the core), it can slide back in.
Reattach using short hose and clamps.

Good luck

update
 

So, yes, after a really short time, the heat began to fade away again, so now I'm interpreting this as the core is shot and clogs easily and needs replacing. So I do plan to do that, and when I do, God willing, I plan to dissect the old core and see what is going on in there and maybe even check a few things out about the water flow direction and come back with some info on it. But I don't plan to do it for awhile as in California, no heat needed now for 6 months or so, but I do pland to do it within the 6 months.

Also, yes, I've seen the way to do it by cutting the lines. Thanks

As far as I know, heater cores are not uni-directional.
They flow in both directions.
This is because mechanics reverse flush the heater cores.
You heater core may have been very clogged.
And I wouldn't waste time on dissecting it
Unless you actually want to see the insides.

Good luck