It sounds like a tape head or mechanism "contamination " problem
I had this exact model ( and a 500) which I happily used with a mixture of tapes .. Ended up with similar problem , not amenable to tape head cleaner.. Had to "write off" the camcorder in the end
It seems to be to much of a coincidence
In my case I had been Using Sony tapes then used Panasonic then a JVC: then the problem arose
Although the advice is to avoid mixing tapes I find it very odd that this is happening after only one tape
Also I had felt it was Sony vs the rest
I still use Sony ..but I stick to Sony
It may not cost much to clean the head professionally. The 350 is a good camcorder and apart from not being especially low light friendly ( a bit of grain) is actually much better than many of today's MiniDV models having a host of manual controls ect
...They don't make them like that anymore