The valves on your old horn may be out of alignment and/or leaking. Those two things cause all kinds of playability issues on old horns.
If you're looking for a cornet-like horn, I'd just buy another cornet - a newer one....
Those cheaper pocket trumpets may disappoint you.
I'd never try to play a 100 year old horn with any regularity unless I were doing some sort of "historical" or period performance. Most of them just don't play very well, even when they are in good condition. Many times, used Yamaha 2330 short cornets can be found for $200 -$300, and are very good horns for the money. Pair it with a Wick or other suitable "real" cornet mouthpiece and you're on your way.