### Summary

The code that checks the connection + become details in a loop is not applying the become state on connections that have the `connected` attribute set to `True`. If connected is `True` the else block will just re-use the existing connection plugin which won't have the `become` details from the first task on it.

https://github.com/ansible/ansible/blob/27ce607a144917e6b9a453813a7df6bbc9ea2213/lib/ansible/executor/task_executor.py#L562-L572

Note there are some other bugs that mask this problem:

* ssh never sets `connected = True` so will always re-create the connection and subsequently the updated become info
* Using `ansible_connection=paramiko` also works because it never matches the connection `_load_name` of `paramiko_ssh`
* Redirected collections will most likely also mask the problem because of the load_name vs `ansible_connection` value check which is most certainly wrong.

The code needs to be updated to do a proper check for the connection name `self._connection._load_name != current_connection` to actually work properly with redirected plugins and aliased names like paramiko and paramiko_ssh. The code also needs to be updated so that if a connection is re-used then the become plugin needs to be added/deleted if necessary.

The connection should also be closed when being dropped as well.

### Issue Type

Bug Report

### Component Name

task_executor
connection
become

### Ansible Version

```console
$ ansible --version
ansible [core 2.14.0.dev0] (low-level-become e91a069260) last updated 2022/08/03 08:18:13 (GMT +1000)
  config file = None
  configured module search path = ['/home/jborean/.ansible/plugins/modules', '/usr/share/ansible/plugins/modules']
  ansible python module location = /home/jborean/dev/ansible/lib/ansible
  ansible collection location = /home/jborean/dev:/home/jborean/ansible/collections:/usr/share/ansible/collections
  executable location = /home/jborean/.pyenv/versions/ansible-310/bin/ansible
  python version = 3.10.2 (main, Jan 18 2022, 12:56:09) [GCC 11.2.1 20211203 (Red Hat 11.2.1-7)] (/home/jborean/.pyenv/versions/3.10.2/envs/ansible-310/bin/python3.10)
  jinja version = 3.1.1
  libyaml = True
```


### Configuration

```console
N/A
```


### OS / Environment

N/A

### Steps to Reproduce

```yaml
- command: whoami
  become: '{{ item }}'
  with_items:
  - true
  - false
```

Doing `false` first reversed the problem where the 2nd iteration will not run with become

### Expected Results

First task is run with become and the 2nd is run without.

### Actual Results

```console
Both tasks are run with the first `become` value set.
```


### Code of Conduct

- [X] I agree to follow the Ansible Code of Conduct
item_ignore = item.pop('_ansible_ignore_errors')
                            if not res.get('failed'):
                                res['failed'] = True
                                res['msg'] = 'One or more items failed'
                                self._task.ignore_errors = item_ignore
                            elif self._task.ignore_errors and not item_ignore:
                                self._task.ignore_errors = item_ignore
                        
                        for array in ['warnings', 'deprecations']:
                            if array in item and item[array]:
                                if array not in res:
                                    res[array] = []
                                if not isinstance(item[array], list):
                                    item[array] = [item[array]]
                                res[array] = res[array] + item[array]
                                del item[array]
                    if not res.get('failed', False):
                        res['msg'] = 'All items completed'
                    if res['skipped']:
                        res['msg'] = 'All items skipped'
                else:
                    res = dict(changed=False, skipped=True, skipped_reason='No items in the list', results=[])
            else:
                display.debug("calling self._execute()")
                res = self._execute()
                display.debug("_execute() done")
            
            if 'changed' not in res:
                res['changed'] = False
            def _clean_res(res, errors='surrogate_or_strict'):