As you said you were on a budget I have to recommend a good sand and re-paint over the replacement option. If you take the time to do it properly it will look great, but it will take time and elbow grease as Seaview said!
Start with a rough-ish paper (120 grit if its just thick paint causing the rough finish, or perhaps 80 grit if the paint has particles of dust/grit in it causing the roughness).
Dust it all down (or hoover ideally), this is very important as getting dust in the paint job at any point defeats the entire object! Then paint with a good solvent undercoat to give you a nice smooth base.
Sand again when undercoat is dry with fine paper (180-240 grit) to remove the last imperfections, and dust down/ hoover one final time.
Now you're ready for your top coat of gloss / satin or whatever and everything should be perfectly smooth.
If you want to go one step further, sand again with fine paper and paint an additional coat of gloss / satin but this is rarely needed if previous coats are done properly.
Alternatively, if the old paint is really thick you could strip the old paint back with Nitromorse or some other stripping compound before starting the other steps. This is cheap but even more time consuming and can be pretty messy.
Hope this helps