So we have a 846 code of 05/04/2022 date but we are a weekly update code. So my question is is the deposit date accurate and the fees are taken out of our refund?

Well, the IRS can always change the deposit date.  Have you checked the IRS Where's My Refund website?

Once it is actually issued, it can take a day or two for the TT fees to be taken out, and the another 1-5 days for your bank to credit the deposit.