Barrel Trimmers: Several differing opinions on their use. The barrel trimmer must be SHARP, or it will burn the wood as you noted. I have had three with two being carbide. I gave them away and they are still being used. However, there are a couple of ways to trim the end flush and square with sanding jigs. I have what is called a "Sanding Mill". Some have a similar jig for sanding it flush by using the lathe with a sanding disk and jig on it. MUCH better control through sanding.
IF you do want to use a barrel trimmer, there are two that are head and shoulders above the rest: Woodpeckers barrel trimmer and Whiteside barrel trimmer. Both have carbide cutters and are exceptionally sharp and have a price to match. These two companies make very high quality products.
AS to putting blanks on the mandrel, you mentioned - a thicker tube. I wouldn't know about that, but it has been quite common for those new to turning to get glue inside the tube and that causes considerable trouble down the road. I'm not saying that you have glue inside the tube but I ask to please double check that.