Would passing this constitutional amendment open up the constitution for other changes or lead to a constitutional convention?

No. Passing CACR 22 will only add victims’ rights to the constitution, it won’t take away anything already enshrined in the constitution or create an opportunity for content to be added or changed. 
The New Hampshire Constitution has been amended 200 times. NH has two methods to amend the state constitution: 1) through a legislatively referred amendment, like CACR 22; 2) a constitutional convention. A legislatively referred amendment moves through the legislature just like any other bill, it just has a higher threshold for passage, 3/5 of the NH General Court, then it would be added to the ballot and requires support of 2/3 of the electorate. This process does not open up the constitution in any way.