On June 26, Connecticut Governor M. Jodi Rell signed SB 913. It provides that when a vacancy occurs in a U.S. Senate seat, the vacancy should be filled with a special election (unless the vacancy is quite close in time to a regularly-scheduled election). Depending on the timing of the vacancy, there might or might not be primaries. If there were no time for a primary, qualified major parties would nominate by convention.
In most states, and in Connecticut until now, vacancies in U.S. Senate seats are filled by the Governor, although the appointee only serves until the next regularly-scheduled congressional election.
Looking through the new law I do not see anything about minor party or independents nominating candidates. Is there old language that still applies to this?
Yes, the old language would cover them. There are already Connecticut laws for special elections (for example, for US House).
What states now fill U. S. Senate vacancies other than by gubernatorial appointment?
#3 The 17th Amendment requires an election, but in most cases the special election is delayed until the time of regular congressional elections.
Texas requires the special election to be held at the next uniform election date (in May or November now, there used to be 4), though the governor has the authority to set an earlier date.
Congress has the authority to regulate the time and manner of senatorial elections, and did so even when the elections were by the State legislatures.
Other states (besides Connecticut and Texas) that provide that special U.S. Senate elections to fill vacancies should be filled quite soon after the vacancy (without necessarily waiting for the next congressional election) are Alaska, Massachusetts, Oklahoma, Oregon, and Wisconsin.