Welcome to the site, Apkfla85!
The Conaxsat CNX Mini is DVB-S only, so yes, you'd need a new receiver. One that can receive DVB-S2, or else you'll not be able to receive a lot of channels.
As Ke4est mentioned in the post above, the CNX Mini was a popular hack box and we don't discuss that stuff, or have any tolerance for it at all here on this site. But, if you're interested in true FTA, then we can help you here. I'll assume that you're talking about a Ku setup here and not C-band, because you said your dish is up on your roof!
Definitely get a new receiver and probably you'd be good to go. For true FTA you need at least a 30'' dish, [76 cm], but a 36'' is recommended, [90 cm]. Under 36'' and some channels will be hard to receive. For an LNB, a linear with a LO of 10750 or a universal with a LO of 10600 will work. If you do have to buy a different LNB, the PLL ones work real well, Titanium Satellite sells the PLL-1KS for $17 and has great reviews, another good PLL is the Avenger PLL322-S2 or PLL321-S2. [322 is two port, 321 is one.]