The possibilities are a Gite, a B&B, or a hotel.

A gite is a furnished rental that generally is available on a weekly basis and allows one to cook and relax and sometimes swim or enjoy the countryside or the city.  It can be a thousand different things – a fishing cabin, an apartment in a Chateau, an old farm, a classic Maison de Maître.  And it can be super cheap or quite expensive.  In any case, it’s a perfect solution, at least for me.  I don’t need to dress nicely, I can drive out every day and return home when I’m tired, I can cook all that lovely food I’ve bought at the market, I can drink without driving… all of that – and more.  Plus there is usually someone around to answer questions, offer advice, or get the internet going.  Gites can be found both in the countryside or in town.

I've stayed at dozens, and what I do is go online using HomeAway.com or Booking.com  or even just Google.  Search 'Gite Lectoure' - the gites will come up and you will have a selection.  If you don't select a Gite that is in Lectoure, note how far the gite is from Lectoure.  If you use Googlemaps, it will tell you the drive time.  I would not stay more than 1/2 an hour away.

I have not used AirBnB and I don't recommend Gite de France.
I pick solely on location and photos.  If you aren't able to book the dates you want or you you're on the cusp of a price change, don't hesitate to e-mail the owners.  I have asked to have a winter price extended and the owners have agreed.  I have also asked to check in on an off day and the owners have agreed.  These people definitely want to please you - they are usually generous and hospitable.

There will be a list of amenities.  The three you want to check on are: wifi, linens, and heat. I prefer stay at a place that does not charge extra for those items.  If it's summer, you may also want to opt for a pool.  

You should make arrangements to pay your damage deposit on arrival and in cash so that it can be returned to you at the end of your stay.  It isn't easy for gite owners to wire money back to you and amounts may be lost along the way.  Confirm this by e-mail with the owner.

How does a Gite differ from Air BnB?  Well it's possible that there is little difference.  But gites in France are generally holiday lodgings reserved for that purpose.  I avoid staying somewhere where I think I will have to work around the owner's possessions.

Gites can be basic or luxurious.  Look at the photos carefully as price is not always indicative.  I've stayed at a gite with a La Cornue Chateau Range, and one with spectacular antiques. Luxurious gites are usually expat retirement homes abandoned for various reasons to gite-hood. You will find that most Gites are owned by the British.

B&B

A B&B is similar to a Gite, but it comes with less room, less privacy, and rarely the opportunity to cook.  But, it also comes with a host, and breakfast.  It's a cross between a hotel and a Gite. I've also stayed at a number of B&Bs, in part, because I wanted to be able to ask questions about the region.  B&B owners can be incredibly helpful.  Also, B&Bs can usually be booked by the night.

On the other hand, you share the public spaces, in rare cases the bathroom is not 'ensuite', the owners are on-hand, and you can't be as messy.  For those who don't want to be entirely on their own, this might be the best option.

HOTEL
You are familiar with hotels already.  If you decide to stay in a hotel, your best option will be to stay in Lectoure. There is a lovely old, said to be fading, hotel with an excellent  restaurant -The Hotel de Bastard.  There is also a beautiful luxury hotel with 5 suites,  - l'Hotel particulier Guihon.  The Hotel College de Doctrinaire, intended for people enjoying the health spa across the street has been renovated in the last few years, and is very attractive, at least on the outside.  All 3 are less than a 5 minute walk from my home.