Electronic – Signal noise cancelled by touching the leads

You problem probably is with the earth reference between the scope and your circuit. The oscilloscope ground lead is always referenced to the earth connection of your AC plug (except battery operated oscilloscopes). Are you sure that the DC power supply you are using to power your circuit is well referenced to the same earth connection? That does not only mean that the DC power supply ground is connected to earth pin of your AC plug, it also means there is actually a connection between the DC power supply earth to the oscilloscope earth. If it does not, you will probably get the noise you are experiencing.

Continuous mode is probably just automatically repeating pulse mode. It is much easier to design ultrasonic sensor which alternates between transmission and reception and measures time delay before hearing its own pulse, than any continuously transmitting sensor. And frequent enough pulses will work for most applications.

The 100 Hz frequency could be simply the repeating frequency of ultrasonic pulses sent (it gives 3.3 m roundtrip distance, maybe enough for unwanted reflections to be attenuated compared to the specified 25 cm max sensing distance). It is possible that power rail voltage drops or peaks on transmitter start/stop because of extra load and internal regulator for analog output is unable to filter it well enough. You can try to add more capacitance to the power rail close to the sensor. If your sensor and ADC has common ground and single-ended sensing, it can be ground current which makes this peak too. Make sure, there is no ground current flowing between ADC and sensor when it switches on transmitter. Or use a differential measurement.

100 Hz could be some coupling of mains too, but the frequency is bit off (mains frequency is usually pretty exact). But you can easily check if your pulses are in sync with the mains.