Don't quit unless and until you have a signed offer from another company in your pocket.
As you experienced it's hard to get anything unless you have a lot of experience, especially in the web area.
If I were you I'd put that Cobol knowledge to good use. Get training in other enterprise technologies that interest you and start trying to get a job using those (maybe even jobs moving Cobol applications to other platforms like
J2EE).
That way you can use your 7 years in the business as a plus, rather than having to start from scratch as a junior at an age when you are due a senior salary and benefits.
Your work conditions and pay development seem good though.
In the last 5 years I've not had a single increase that was more than compensation for inflation (and usually less than that).
In fact, given that I lost my job in 2003 due to a bankruptcy, I now make (before taxes) a bit less than I did (nominally, I never saw most of it due to the financial problems of that company) in 2002.
btw, I can understand you wanting a change from Cobol. I did it for 18 months and got out because I wanted something more interesting.
If I were you I'd take this up with HR at your company. They may have an internal opening you could move into on another project using other technologies and may well rather pay for crosstraining than see you leave with all the knowledge and experience you've built up over the years.