Do not start looking for anything until you are in cotract.
I once sold a house at the broker's open in June. The contract that was written stated a closing of January 30th or earlier. The seller found a home and and both closed in October.
If the notation is in the listing, someone who needs a home immediately will know this is not for them.
In a very brisk market like this, sale contingencies and reverse contingencies are not well received by buyers or sellers.
If you want to move and can live with the possibility of selling, closing and moving to temporary housing until the right place for you comes along for you, then do just that.
Of course you can. You can place any contingencies you want to - it's your house. However, most buyers will not want to do that so you may not get any offers. You can try it if it's necessary for you - after all, it only takes one buyer to really want your house and they may be willing to wait?