In my humble opinion, yes, you should. Your a captive audience for the agent who is going to get both sides of your business. In our market it's not uncommon for brokers to offer "move up" programs where they will discount the listing fee if you agree to use them to buy your new home. Since the broker will typically make a full buyer's agent commission on the buy side, they often discount their portion of the listing fee in this type of a situation. But remember, even if they typically charge say 6% for the listing commission, some portion, typically half, is going to get paid to the agent representing the buyer, so they can't just give up the full 6%.
Presuming the buyer's agent norm in your market is 3% and the typical listing commission is 6%, leaving 3% for the listing agent to earn on the sale of your house, and then another 3% for them to earn when you buy your new home, it's not uncommon to see a discount off the listing agent commission of half, or 1 to 1.5%. This is fair and reasonable for your agent. They will make 4.5% to 5% total.
It's not a bad idea to go hunting around your market by looking at broker websites and seeing if any of them have these "move up" type programs. Call several brokers and ask them what their policy is. After you do your research, present a recommendation to your agent and work out something that is amicable and fair.