What do you mean it doesn't display full screen? is it cut off? the Sony Trinitron should be equipped with a S-Video input depending on your model? If so you could always buy a Dramcast S-video or consider getting a VGA adapter and using a PC monitor. in either case you will get a better picture and might not need to buy a tv.
If you do buy a tv its hard to recommend something, if you want something small like that 13 inch anything with composite inputs should work as the picture is so small you will not see as large of improvement as you would using a good cable type and tv. I use the VGA box and a VGA to component converter up scaling box with a few different flat panel HDtv and it works pretty good. I payed about $50 for the box a while back and I think they are cheaper now and I think that the new ones have HDMI out instead of component. The one I use only displays in 720p and 1080i max and the new ones should do 1080p. The VGA boxes are pretty cheap too somewhere around $15
here are a few ebay listings that might work, I have not personally used either of these but they seem like they would work. I would actually like to get one of these for my self, but it seems like a waste already having both in a different version. Also keep in mind that not every Dreamcast game supports VGA, but most can be sued with it in some form.
DC VGA cord roughly $8
http://www.ebay.com/itm/VGA-High-Defini ... Sw1DtXLDc8Converter box $13.99
http://www.ebay.com/itm/VGA-to-HDMI-Swi ... SwzvlW-21T