With your budget, you would probably have to pick up something like this. Personally, I'd go with a brand name burner at about twice the price, but this might make an intermediate step. You can also save up so you could afford something like the Colwood Detailer.
No, You need thinner tips. That's one of the reasons the higher end burners use wire or razor thin tips - they heat up very quickly.
Any time a tip contacts any surface, it will transfer heat, and cool down. The longer it takes that tip to re-heat, the more uneven the burn will be. Another thing to consider, is the medium that's being burned. Wood similar to Pine has an uneven grain - tighter, harder growth rings in the winter, and looser in the summer. The loose grain burns easily and the winter grain is harder to burn.
If you want a cheap solution to see if you like the hobby, you could buy something like this. It's a cheap Chinese burner, that may or may not have reliability or longevity issues, but many users seem to like them. If you're sure that you want to take up this hobby, a better solution would be the Colwood Detailer. I've had one for years and love it. You can buy additional tips as you feel the need.