Single Supply AC Gain Opamp With Unity DC Gain
A basic problem with attaching oscilloscope leads to high impedance nodes is that the input impedance of the o-scope circuits are loading those nodes. O-scopes typically have an input impedance of around 1 Mohm and, with an effective impedance of 2 Mohms || 2 Mohms your bias voltage of 1.65 volts is going to approximately halve to around 0.8 volts. If your o-scope probes have the option of a 10:1 operation, the effective impedance becomes more like 10 Mohms so, this would be the preference.

Another observation: you do need to have a supply bypass capacitor but your circuit doesn't show one. Data sheet extract: -


  Connect low-ESR, 0.1-µF ceramic bypass capacitors between each supply
  pin and ground, placed as close to the device as possible. A single
  bypass capacitor from V+ to ground is adequate for single-supply
  applications


Next....


  Once the output had flat lined I noticed the current being pulled by
  the opamp was around 50mA!!


This may be a fault on your circuit pulling down the 3.3 volt supply rail - please check to see at what point the output started to flat-line and confirm if it coincided with the draw of 50 mA or whether the 50 mA built up gradually over the period. As a side note, the TLV9062 is capable of outputting 50 mA into a short circuit so maybe there was an excessive load on the output?

You say you are using a TLV9062 but some of those parts come with a shutdown pin and it's not clear from your question whether you are using one of those parts and, if you are, how you are wiring the shutdown pin.


  To ensure optimal shutdown behavior, the SHDN pins should be driven
  with valid logic signals. A valid logic low is defined as a voltage
  between V– and V– + 0.2 V. A valid logic high is defined as a voltage
  between V– + 1.2 V and V+. The shutdown pin must either be connected
  to a valid high or a low voltage or driven, and not left as an open
  circuit. There is no internal pull-up to enable the amplifier.


And, both shutdown pins need to be properly controlled to avoid problems even when only using one amplifier in the package. The other amplifier (if unused) should have applied sensible levels to its inputs as a precaution.


  I was hoping for some direction on this issue. I have no idea where to
  start.


Some direction given!