Finding zero cross of AC signal digitally
According to your comments, you don't want to find the zero crossing per se. You want to find the RMS of the AC, and quickly, within a cycle or so (put these more specific requirements into your question). No need for extreme accuracy? The method you choose depends heavily on what accuracy you require, you can get away with all sorts of things at 10% that you could not at 0.1%.
It's neither necessary nor sufficient to have the zero crossing available. What is required is to sample exactly one cycle of the mains waveform, if you want to do it in one cycle, or at least 5 cycles without being synchronous, if you can tolerate the latency and want to target 0.1%.
The errors for sampling slightly more or less than one cycle are small, and if you don't want to sync up your sampling rate to exactly an integer multiple of the mains frequency by PLL (as the mains frequency can vary a little), then capturing just over 1.5 cycles allows you to select an entire cycle between approximately zero crossings.
Once you have a whole cycle, then the DC value is just the average of those samples.
It's well worth cranking up an excel spreadsheet to experiment to see what sort of errors you get with a sampling rate deviating from exactly an integer multiple of mains, and start and finish deviating from zero crossings.
Note that if you measure the RMS, and the measurement is in error because of the DC term, then that error is due to the RMS sum of those. For instance, the following 8x sampling 100,71,0,-71,-100,-71,0,71 gives an rms of 70.86, whereas 110,81,10,-61,-90,-61,10,81 (a DC offset of 10% of the peak) gives an rms of 71.47, an error of 0.8%. That took me 60 seconds to do on a spreadsheet. Now do the same for yourself, but varying the accuracy of the integer oversampling, and starting/stopping at near or far from zero, and compare the accuracy with what you want.
You may want to think about what sort of rate you capture at. You suggest 8kHz, which is a lot of processing, and not an exact multiple of 50 or 60 Hz. A multiple of 300 Hz gives you an exact multiple of both. How many samples do you want? It depends on the harmonics, and the accuracy you want in their presence. 1200 Hz is adequate for most commercial metering applications. Fewer samples mean you have more time too do work with them.
I can suggest a specific algorithm. Set your sampling rate to 1200 Hz. When you first switch on, do an initial calibration to find out whether you're in 50 Hz or 60 Hz land. Then record exactly 24 or 20 samples respectively, which will occupy exactly one cycle. Average them to get the DC value, and subtract it. Then square, sum, and divide by 24 or 20 (which means multiply by the reciprocal of 24 or 20, you don't need to divide by a constant). You may if you want assume that the DC level is either going to remain consistent, or if it does vary, shift very slowly. You can then, for minimum latency, switch to capturing half cycles of 12 or 10 samples, or even quarter cycles (though for 60 Hz you'd have to increase the sampling rate to 2400, with the corresponding other adjustments, to get an integer number of samples in a quarter cycle), and use the assumed value of DC. If needed, keep track of the current DC level with a very slow filter, updated every time you take a half or quarter cycle.
Of course, if your hardware can set the sampling rate, then there's no need to pick something fixed like 1200 to handle 50/60. Once you've discovered which it is, you can set the sampling rate to 16x say, and always take and scale by 16 samples, which of course just means a shift of 4 places. Note that while recording a whole cycle will reject all harmonics, taking a half cycle will now not reject even harmonics, for instance a half-wave rectified load, which are fairly unlikely to occur. A half cycle will still handle the flat-topped distortion due to a (very common) rectifier+capacitor load.