Electronic – How to reshape this PCB antenna so that I can still use it with the given circuit

The big question is: what distance do you want to cover? The data sheet of the transmitter quotes a maximum range of 50 metres [about 150 ft]. Will you use that, or will the receiver be closer?

Any oscillating signal will radiate: the whole point of the USA's FCC is to limit the amount of annoying [or dangerous] EM radiation coming from devices. Depending on the range, and the presence of bulky metallic objects between transmit and receive, Your Mileage May Vary.

Antenna theory and design can be taught in depth by amateur radio enthusiasts, or groups like ARRL. For starters, a simple piece of wire about 20cm long can act like a "whip" antenna: keep it clear of grounded cases, and it should be enough to get you started. A second piece the same length can act as your receive antenna. Start with the circuits next to each other, make sure they work, then seperate them. If they stop working before reaching the seperation you want, THEN [and only then!] consider delving into antenna theory...

If you need more, I'd suggest TI's Application Note #AN058: http://www.ti.com/lit/an/swra161b/swra161b.pdf

And: http://www.picaxe.orconhosting.net.nz/yagi433.jpg

The results you have are a bit odd, due to the lack of a ground plane under the antenna (as the datasheet for the antenna suggests http://www.ti.com/lit/an/swru120c/swru120c.pdf) the PCB thickness should not matter, there isnt a reference plane that will modify the antenna characteristics themselves (you did remove the ground plane from underneath the antenna right?)

What does change however is the feed line from your tranciever to the antenna, which will need to be modified to maintain the same impedance as before, as your layer stackup is different.

Also, as SteelRat pointed out in the comments, you might want to use the same copper thickness as the TI stackup, as that has some effect on impedance, he indicated it is the copper thickness that might make or break the antenna. So if you follow TIs recommendations with copper thickness, shape, ground plane recommendations, and modify your feed trace to 50 ohms, then your design is quite likely to be good and stable.