The TSOP4P38 is different from the digital TSOP receivers in that it utilizes suppression of sensitivity (i.e., AGC) of the detector to produce a signal that can be related to the strength of reflection and hence distance.
The schematic I posted is an example of how to use one 555 to modulate the 38 KHz produced by another. In that example, the 38 KHz is produced by IC1A. The modulation is by IC1B. The modulation frequency I used was about 800 Hz, and the duty cycle was about 50%. For the TSOP4P38, you need to use a frequency of about 2 Hz and a duty cycle of 24% (as shown in the example posted above). I suspect the 24% duty cycle is not critical, but it is important to use a low frequency, so the AGC will kick in and reduce gain.
I suggest that you design the transmitter first. You might want to use an MCU to do that, since your project seems to be MCU-based. If you prefer to use just 555's, get a circuit for 38 KHz, which you have. Then get a circuit for 2 Hz with a duty cycle that can be adjusted to less than 50%. There are various ways to do that. I prefer to use diodes to bypass resistors in the charge/discharge part of the circuit. Paisley has suitable circuits on his site:
**broken link removed**
Finally, use the 2 Hz oscillator to modulate the 38 Khz oscillator as in the schematic I posted above. If you have access to an oscilloscope, it will help immensely in getting the timing right.
John