This is a tough decision, but the answer will depend on your personal situation, as well as the condition of the local housing market.
If you put your home on the market first, you may have to scramble to find another one before settlement, which could cause you to buy a home that does not meet all your requirements. If you cannot find another home, you may need to move twice, temporarily staying with relatives or in a hotel.
On the other hand, if you make an offer to buy first, you may be tempted to sell your existing home quickly, even at a lower price.
The advantage of buying first is you can shop carefully for the right home and feel comfortable with your decision before putting the existing home on the market.
On the flip side, the advantage of selling your existing home first is that it maximizes your negotiating position because you are under no pressure to sell quickly. It also eliminates the need to carry two mortgages at once.
Talk with your agent for advice. Discuss the pros and of each and whether certain contingencies written into the contract can ease some of the pressures.