How to assign payment without changing the existing transaction

[yary]

Short version: how exactly do I assign an existing transaction to an
invoice as a payment, without having that original transaction change?

Long version:

I'm trying to assign existing transactions to customer invoices, but doing
so always reassigns the transaction. I suspect I'm trying to assign the
wrong transaction, or that my QIF transaction import wasn't as successful
as I think it is. This is part of converting an existing "book" to GnuCash.
Here's my accounting flow, and what I've tried so far.

"In real life" someone pays by check or cash. In the old accounting system,
I would mark the invoice as paid, transferring it to an "Undeposited Funds"
account. Later I'd go to the bank and deposit the payment, and enter a
transfer from "Undeposited Funds" to "Checking Account"

I used GnuCash's import facilities to re-create my account tree (chart of
accounts), customers, vendors, invoices, and general ledger (all the
transactions). As advised, I'm using "Assign to payment" to tie the payment
transactions to invoices.

1. Assigning the "checking account" transaction. Here's a sample
transaction in the account that could be assigned to an invoice. It's dated
Feb 20 2013 (I know it's old, but I'm also doing this for current items...)
screenshot at
https://www.dropbox.com/s/0fi20r64rvbf8ez/Before-Assign.PNG?dl=0

I can right-click it, assign it to the proper customer & invoice, and the
invoice is marked as paid, BUT then that transaction disappears from the
"Checking Account." That bothers me, because now I've lost the record in
the "Checking Account" of the deposit on Feb 20 2013.

2. Assigning the "Undeposited Funds" transaction. I don't like this option
quite as much, because not all payments go through here- sometimes I would
deposit a payment the same day and avoid this. And it also has similar
issues. Here's a payment from Apr 2 '15
https://www.dropbox.com/s/0vhgyiswt471xeh/Undep-Funds.PNG?dl=0

I can right-click on the line with the $165 deposit and assign it to
invoice #62, but then the deposit switches accounts. Instead of being in
the "Checking Account" it goes to "Accounts Receivable"

(If I assign the top line- the one already marked as A/R as the transfer
account- then the accounts all look good, but it's a refund not a payment!)

3. Looking in the A/R list, there are payment transactions in it which I
can assign- but doing so doesn't mark the invoice as paid. Here's a
screen-shot after assigning:
https://www.dropbox.com/s/z9tqst3g6sx798s/A-R-assign.PNG?dl=0

Not sure how to show that it didn't assign it to the payment I selected, so
here's the lot viewer showing that there are now two lots in the amount of
the invoice- one for the original invoice, one for the payment- with no
"gains" on the invoice.
https://www.dropbox.com/s/ku2nvix6riagf3w/Lots-after-assign-to-invoice62.PNG?dl=0

-y