I would suggest those Wills would be certainly worth obtaining and checking - well done Newlyn.
Now, I have been a little busy this afternoon too and have found a bit of stuff.
Unfortunately I do not have records for Kenwyn but I do have quite a bit for Camborne so here it is.
1641 Protestation Return - CamborneGilbert ACHIM
Camborne Marriages 1538-174513 Nov 1604 [Rob]ert ACHYM m. Christian (suggest this is Gilbert - see later)
07 Aug 1637 Gilbert ACHIM m. Alce HUMFREY
23 Oct 1658 Nicholas WADE m. Alce ACHEM
04 Oct 1666 Laurence ACHEM m. Elizabeth PAULL
08 Nov 1729 Gilbart EACHEM m. Alce BRANCH
(I did not spot this last one until I re-read Newlyn's note and went back to check it. :
Marriages at Perranuthnoe 1589-1812Willyam HILL m. Fraunces d/o John EACHEM
Camborne Baptisms 1558-171915 Jun 1638 Gilbert s/o Gilbert ACHIM
21 Jan 1639 Laurence s/o Gilbert ACHIM
08 Mar 1642 Richard s/o Gilbert ACHIM
07 Jul 1667 Gilberye s/o Laurence ACHEM
10 Oct 1668 Elizabeth d/o Laurence ACHEM
29 May 1671 Alce d/o Laurence ACHEM
15 Jun 1673 Grace d/o Laurence ACHEM
05 Sep 1675 Charity d/o Lawrence ACHEM
16 Sep 1677 John s/o Laurence ACHEM
22 Jun 1679 Richard s/o Laurence ACHIM
02 Feb 1680 Honner d/o Laurence ACHIM
Camborne Burials 1538-174918 Jul 1604 Edmond ACHYM
05 Jun 1616 ACHIM of Gilberti
14 Mar 1626 Johane EACHIM, wid.
13 Feb 1635 Christian ACHIM, w. of Gilbert
09 Jun 1653 Gillbertt ACHEM
30 Sep 1655 Gilbert ACHEM jnr.
25 Jul 1667 Gilberte ACHEM son of Laurence
30 Oct 1677 Charity ACHEM dau of Laurence
25 Feb 1679 Richard ACHIM son of Laurence
09 Apr 1682 Honner ACHIM dau of Laurence
01 Feb 1683 Elizabeth ACHIM dau of Laurence
12 Feb 1711 Elizabeth EACHIM
04 Apr 1743 Gilbert ACHAM
The 1604 burial of Edmonde suggests the link to the Edmond at Sancreed half a century later.
In the early part of the Baptisms there are multitudes of baptisms with the surname either not recorded or indecipherable.
But basically we seem to be looking at one family and it will be interesting to see what can be found at Kenwyn.
I will take a look at the IGI records and see what I find and will also check Camborne Baptisms through until about 1750.
CT