I have had great success with 5 or 10cc syringe, with large gauge plastic needle with a bit of cotton stuffed into the end. (Actually I have just one size of small gauge plastic tips, and I cut them at a slight angle to make a big opening.) Make sure you use the luer lock syringes, so the tips screw on. The cotton doesn't make a "brush," it prevents the flux from squirting out uncontrollably... which will happen because flux is sticky and the syringe will not move perfectly smoothly.
When I first use it for the day, the tip will be hard/dry, particularly if I neglected to put it back in the point-down stand. I just squeeze out a little flux (this is why you want luer lock; you can build a lot of pressure if/when needed) and then touch the tip to something (I have a little board for this) to spread out the flux over the entire tip to get it wet, so it works (flux wicks/spreads/flows over the entire tip, not just exit and beat out of one tiny point; and the hard deposits quickly dissolve).
I usually just throw them out after one use, but you can refill a syringe with rosin flux maybe 3 times before the seal starts to stick to the side of the syringe and go a little wonky. 100 pack of syringes and plastic tips is peanuts.
This syringe can apply a lot thicker of a bead/pool of flux, when desirable. You can also squirt out small amounts into a container or over a little board, if you want to apply it with a stick or a brush.