It is entirely common to accept an offer on your current home before you have found your next one, and it does not need to be as stressful as it might sound.
Selling first has genuine advantages
Once your sale is agreed, or even completed, you become a chain-free buyer, which is often a real advantage when it comes to negotiating on your next property. Sellers generally prefer a buyer who is not waiting on their own sale to fall into place.
A longer completion date can buy you time
It is worth discussing a longer completion period with your buyer if you need more time to find somewhere. Many buyers, particularly those without their own onward chain, are flexible on timing if it means securing the property.
Renting between moves is a reasonable bridge
If the right property has not turned up by the time you need to move, a short-term rental is a genuinely reasonable option, giving you a base to search from without the pressure of a fixed deadline hanging over every viewing.
Storage can bridge the gap too
If you are between homes, short-term storage for furniture and belongings can make a rental period considerably simpler, since you are not trying to fit a full household's worth of things into a temporary space.
Proceeds from your sale can strengthen your position
If you no longer need a mortgage on your next home, having your funds ready and confirmed can make you a genuinely attractive buyer, sometimes able to move faster than someone still waiting on a chain of their own.
There is no need to rush the decision
Finding the right smaller home is worth taking seriously, and there is no requirement to settle for the first reasonable option simply because your current sale is moving quickly.
Your local associate can talk through which of these options would suit your circumstances best.