Hi Rick,
Here's what I get for not looking at GCA for several days!
I'm a member at Tedesco so that's an option, although last year we only had like 7 holes open until about 4/20.
Bass Rocks is fun for sure...no pre-conceived notions as it's funky in spots, but in a good old-world way. That part of the North Shore sticks out into the ocean, so mid-April will be brisk, maybe 10-15 degrees colder than downtown Boston.
As for local publics that are hidden gems, there are a bunch, but mid-April will still be "grow away the winter kill" time...you'll still see the marks from the greens covers, maybe some punch holes from last fall's aeration...and greens will be furry.
Beverly Golf & Tennis, Gannon Municipal, and Olde Salem Greens (9) are all Stiles courses worthy of a play if in the area for a bit of time. Further afield you have Red Tail (might not be open that early), the Boston munis George Wright and Franklin Park are fun but will be sketchy early...but they are sometimes scruffy at the best of times so that may be ok. Plymouth has lots of public golf (plus private PCC, which is great!) and that area will be in full swing by mid-April but just that much South will be aeration time...usually mid-march to mid-april south of boston, and early May in town and north. If committing to that area, South Shore CC is the affordable and really cool public between Boston and Plymouth...another one by Stiles.
Cape Arundel is fun too but a good 2 hours from Salem, and you're going the wrong direction for warmth and good golf conditions!
If you are a rater-type access will not be easy as they generally will not let you on that early.