Hearing a short burst of static (called the squelch tail) after un-keying the transmitter is normal, and a sign you have keyed up the repeater.
The bold section is the key point. You are probably opening the repeater.
1. Re-verify tones and offsets. (which sound like they are set right)
2. Get a second radio,to monitor or a cheap SDR receiver setup (monitor repeater output)
The simplest, throw your call sign out with a short message on the repeaters every so often. If someone answers you have your answer.
Another thing to think about if you are going to use CW a lot is to get a CW weighted vanity call sign. It will allow you to choose a call sign to apply for that is easier when using CW. https://fistsna.org/callweight.php There is also an option in the FCC ULS when applying for vanities to apply a CW weight to selections.
You actually are in hotbed of potential HAM activity due to all the parks. You actually have quite a few repeaters, try programming them and just set the radio to scan. BTW, there is a big difference with an HT (even with a 42" Abree antenna), a 1/4 mag mount attached to a car roof connected to your HT, and a base station antenna llike a Diamond X300. A lot of it has to do with height but the ground plane also makes a big difference.
Here is a fun project, make a tape measure yagi, connect it to the HT. Start pointing it at the repeater general locations and try again from the roof. Chances are the repeaters are on top of the mountains, Worst case you spent $10 and now have a yagi you can use for Fox hunts and satellite work (Which I think you'll need a new HT since the downlink is 70cm, might be able to repurpose some GMRS gear for receive only for that ).