Yes, it is possible and very common. Choose each agent for the job. Selling your house is a huge job and very expensive proposition for you. Only go with an agent after careful interviews and consideration.
If you need to sell your house before buying, I would do the selling first.
When selling your primary exposure is the MLS.
Spend time now reviewing each room for fixes and your personal plan for marketing your house. When you've got the house ready, contact several agents, interviewing a few before deciding on one.
Your first job is marketing your house to the agents.

But you might consider negotiating a better commission rate by agreeing to both sell and buy using the same agent. That's quite common.

I doubt if it's very common unless the Vendor is turning around and buying from a new home builder that doesn't cooperate with agents.

I was a broker for many years and I can't think of a greater insult than for a client to think I was only fit for handling one end of the transaction. On the other hand, the most successful agents I know are the ones who concentrate on selling. When you have a listing, you've got something. Chauffeuring around prospects is no guarantee of anything. Though maybe it's a little better now where there are specifically buyer's agents.

Since I left the business--allergies made it impossible to continue--I've stuck with one broker through close to 30 sales and purchases. I definitely get a better commission rate, and instant service, than I would shopping around.

I doubt if it's very common

I was referring to the OP, not the post directly prior.

I was a broker for many years and I can't think of a greater insult than for a client to think I was only fit for handling one end of the transaction.
The goal is for the OP to sell her home and buy another.
Why the drama about being insulted ? If the person is not the right one, find the right one. If this is your biggest insult, you've led a very great life..

But you might consider negotiating a better commission rate by agreeing to both sell and buy using the same agent. That's quite common.

Saving a little commission is not worth going with the wrong agent.
This is a tough market, find the best agent for each job, sometimes that means two different agents.

Of course, it's not about the broker's or agent's feelings! I fully understand the attitude of most "consumers". Agents in the business for any length of time develop a hard shell to deflect the knives aimed for their backs.

It's just that I fail to see why an agent capable of finding and negotiating a purchase is not capable of selling your home. (If you had a bad experience with the purchase, I can understand splitting the jobs.) I'm assuming this is an MLS area, where all agents have instant information and access to every property.

We chose an exclusive buyers agent for our home & lot search - and when that was completed and it was time to sell our home we asked his advice, interviewed several selling agents, and picked the one agent who exclusively did seller listings for her office.

She had the same standard commissions to all the others - one agent offered a much lower commission if he brokered both sides, which I wasn't comfortable with. She also offered more services and included staging etc...

An interesting side note is she come up with the lowest list price of all the agents we interviewed...