Electrical – Opamp Vos trimming in a single supply circuit

This certainly sounds like instability. Is that opamp "unity gain stable". If not, that's the problem, and that's likely what is happening. If so, you have to use a different opamp if you really need unity gain.

Depending on the opamp and what you really need this circuit to do, you could try giving it some gain. To do that, put a resistor between the output and the - input, then a resistor and capacitor in series from there to ground. The resistor and capacitor will form a high pass filter. Your input coupling already high pass filters with a rolloff of about 1.8 MHz, so you can set the other filter a octave or two below that so as to not matter.

Added in response to datasheet:

You realize this is a video amp, right? You might have mentioned that earlier. High speed amps like this tend to be a finnicky, and you have to pay special attention to layout. You want a nice ground plane, careful decoupling, etc. This is a case where you really need a proper PC board or use RF breadbording techniques, no solderless plug-in breadboard.

The datasheet suggests that this amp should be stable at unity gain, but doesn't come out and actually say so that I found from a quick skim. You would normally use this kind of amp with a gain of 2 at least, like they do in the example on page 13. At these frequencies, you have to consider transmission line effects, so you are usually down by a factor of two due to driving thru a resistor, which then forms a voltage divider with the characteristic impedance. They aren't really expecting anyone to use this amp at a gain of 1.

If the signal out of the dual-supply op-amps is also bipolar around the virtual ground rail, the single-supply parts will face challenges with such signal - unless clipping at the virtual ground rail is the design intent.

Assuming that all signal in the single-supply op amp portion of the design is definitely above the virtual ground potential, the "new stage" does not use some internal-to-the-chip negative rail, it uses the virtual ground as its most negative supply rail.

One point of consideration for such designs is to ensure that the virtual ground itself is a very low-impedance rail with sufficient current sourcing and sinking capacity. Use an op-amp based rail splitter with hefty decoupling capacitors between all rails, and perhaps add a BJT stage, to achieve this.

Alternatively, use a purpose-built rail splitter part such as the Texas Instruments TLE2426 for the virtual ground.

Also note that your single-supply stage output will be bound between Virtual Ground and the positive rail, not utilizing the lower half of the possible voltage range. In other words, output signal peak to peak is limited to ~4.5 Volts (assuming RRO OpAmps) and not ~9 Volts.