If both the homes are in nearby neighborhoods and you can find an agent who has a strong knowledge of both markets, then yes, you should absolutely use the same agent. Working with one agent will make it much easier to coordinate the process especially since you have to sell your home in order to get the money to buy a new one. There are a lot of details to consider and it’s much easier if one agent is managing everything so that nothing falls through the cracks.
Now, the type of market you’re in will determine the best way to approach your sale and purchase. And what I mean by this is that if you’re in a seller’s market where there is low inventory and high demand (buyers competing over properties) then you may have a tough time getting your offer accepted if it’s contingent on selling your home. The seller will compare your offer to another offer that is not contingent; it will be an easy decision because the other buyer promises an easier transaction with fewer potential hurdles and hangups. On the other hand, if buyers are in control in your local market – there’s a lot of inventory, and little demand – then a seller may be more willing to accept your contingent offer and patiently wait while you get your house on the market and sell it.
Since you have already identified your upleg (new home purchase), you should at least get your house on the market quickly, and make an offer on the other home. Have your agent explain to the listing agent that you are aggressively marketing your home to find a buyer. You should build into your contract a timeframe that allows you to market your home while the seller of your upleg pulls their property off the market. I normally have my clients ask for 30 days. This means that you would have 30 days to market your home and get a buyer. Then ideally, the escrow period (this is what we call it in California) for both properties will run concurrently and you can close the sale of your home one day before the sale of your new home. (This may differ a little bit in your state depending on your closing process.)
If the seller does not accept your offer and you still want to move (and you are in a seller’s market), then I would get your house on the market and sell your house contingent on you finding a new upleg.